ResearchArticle High-Speed Transmission in Long …downloads.hindawi.com/journals/ijde/2018/8236942.pdfHigh-Speed Transmission in Long-Haul Electrical Systems BeatrizJuárez-Campos,1

Research ArticleHigh-Speed Transmission in Long-Haul Electrical Systems

Beatriz Juaacuterez-Campos1 Elena I Kaikina 2

Pavel I Naumkin 2 and Heacutector Francisco Ruiz-Paredes1

1 Instituto Tecnologico de Morelia Avenida Tecnologico No 1500 Lomas de Santiaguito 58120 Morelia MICH Mexico2Centro de Ciencias Matematicas UNAM Campus Morelia AP 61-3 Xangari 58089 Morelia MICH Mexico

Correspondence should be addressed to Elena I Kaikina ekaikinamatmorunammx

Received 29 January 2018 Accepted 12 March 2018 Published 18 April 2018

Academic Editor Sining Zheng

Copyright copy 2018 Beatriz Juarez-Campos et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We study the equations governing the high-speed transmission in long-haul electrical systems 119894120597119905119906 minus (13)|120597119909|3119906 = 119894120582120597119909(|119906|2119906)(119905 119909) isin R+ timesR 119906(0 119909) = 1199060(119909) 119909 isin R where 120582 isin R |120597119909|120572 = Fminus1|120585|120572F andF is the Fourier transformation Our purpose in thispaper is to obtain the large time asymptotics for the solutions under the nonzero mass condition int 1199060(119909)119889119909 = 0

1 Introduction

We study the equations governing the high-speed transmis-sion in long-haul electrical systems

119894120597119905119906 minus 13 100381610038161003816100381612059711990910038161003816100381610038163 119906 = 119894120582120597119909 (|119906|2 119906) (119905 119909) isin R+ timesR

119906 (0 119909) = 1199060 (119909) 119909 isin R (1)

where 120582 isin R |120597119909|3 = Fminus1|120585|3F and F is the Fouriertransformation defined byF120601 = (1radic2120587) int

R119890minus119894119909120585120601119889119909 Note

that we have the relation 119906(minus119905 119909) = 119906(119905 minus119909) so we can onlyconsider the case 119905 gt 0 For the regular solution of (1) we havethe conservation law 119906(119905)L2 = 1199060L2 We are interested inthe case of nonzero mass condition int

R1199060(119909)119889119909 = 0 By (1) we

get the conservation of the mass intR119906(119905 119909)119889119909 = int

R1199060(119909)119889119909 =0 for all 119905 gt 0

This equation arises in the context of high-speed solitontransmission in long-haul optical communication system [1]Also it can be considered as a particular form of the higherorder nonlinear Schrodinger equation introduced by [2] todescribe the nonlinear propagation of pulses through opticalfibersThis equation also represents the propagation of pulsesby taking higher dispersion effects into account than thosegiven by the Schrodinger equation (see [3ndash11])

The higher order nonlinear Schrodinger equations havebeen widely studied recently For the local and global well-posedness of the Cauchy problem we refer to [12ndash14] andreferences cited thereinThe dispersive blow-upwas obtainedin [15] The existence and uniqueness of solutions to (1) wereproved in [16ndash25] and the smoothing properties of solutionswere studied in [18ndash21 24 26ndash31] The blow-up effect for aspecial class of slowly decaying solutions of Cauchy problem(1) was found in [32]

As far as we know the question of the large time asymp-totics for solutions to Cauchy problem (1) is an open problemWe develop here the factorization technique originated in ourprevious papers [33ndash38]

We denote the Lebesgue space by L119901 = 120601 isin S1015840 120601L119901 ltinfin where the norm 120601L119901 = (int |120601(119909)|119901119889119909)1119901 for 1 le 119901 ltinfin and 120601Linfin = sup119909isinR|120601(119909)| The weighted Sobolev spaceis H119898119904

119901 = 120593 isin S1015840 120601H119898119904119901 = ⟨119909⟩119904⟨119894120597119909⟩119898120601L119901 lt infinwhere 119898 119904 isin R 1 le 119901 le infin ⟨119909⟩ = radic1 + 1199092 and⟨119894120597119909⟩ = radic1 minus 1205972119909 We also use the notations H119898119904 = H119898119904

2 H119898 = H1198980 shortly if it does not cause any confusionLet C(IB) be the space of continuous functions from aninterval I to a Banach space B Different positive constantsmight be denoted by the same letter 119862 We denote byF120601 or120601(120585) = (1radic2120587) int

R119890minus119894119909120585120601(119909)119889119909 the Fourier transform of the

HindawiInternational Journal of Differential EquationsVolume 2018 Article ID 8236942 13 pageshttpsdoiorg10115520188236942

2 International Journal of Differential Equations

function 120601 then the inverse Fourier transformation is givenbyFminus1120601 = (1radic2120587) int

R119890119894119909120585120601(120585)119889120585

We are now in a position to state our result

Theorem 1 Assume that the initial data 1199060 isin H1 capH01 havea sufficiently small norm 1199060H1capH01 le 120576 Then there exists aunique global solution F119890minus(1198941199053)|120597119909|3119906 isin C([0infin) Linfin cap H01)of Cauchy problem (1) Furthermore the estimate

sup119905gt0

(100381710038171003817100381710038171003817F119890minus(1198941199053)|120597119909|3119906 (119905)100381710038171003817100381710038171003817Linfin+ ⟨119905⟩minus16 100381710038171003817100381710038171003817119909119890minus(1198941199053)|120597119909|3119906 (119905)100381710038171003817100381710038171003817L2+ ⟨119905⟩(13)(1minus1119901) 119906 (119905)L119901) le 119862120576

(2)

is true where 119901 gt 4Next we prove the existence of the self-similar solutions

V119898(119905 119909) = 119905minus13119891119898(119909119905minus13)Theorem 2 There exists a unique solution of Cauchy problem(1) in the self-similar form V119898(119905 119909) = 119905minus13119891119898(119909119905minus13) such that

119898 = 1radic2120587 intR V119898 (119905 119909) 119889119909 = 1radic2120587 intR 119891119898 (119909) 119889119909 = 0 (3)

where119898 is sufficiently small number and

F119890minus(1198941199053)|120597119909|3V119898 isin C ([1infin) Linfin) 119909119890minus(1198941199053)|120597119909|3V119898 isin C ([1infin) L2) (4)

Furthermore the estimate

sup119905gt1

(100381710038171003817100381710038171003817F119890minus(1198941199053)|120597119909|3V119898 (119905)100381710038171003817100381710038171003817Linfin+ 119905minus16 100381710038171003817100381710038171003817119909119890minus(1198941199053)|120597119909|3V119898 (119905)100381710038171003817100381710038171003817L2+ 119905(13)(1minus1119901) 1003817100381710038171003817V119898 (119905)1003817100381710038171003817L119901) le 119862 |119898|

(5)

is true where 119901 gt 4Nowwe state the stability of solutions to Cauchy problem

(1) in the neighborhood of the self-similar solution V119898(119905 119909)Theorem 3 Suppose that

1radic2120587 intR 119891119898 (119909) 119889119909 = 1radic2120587 intR 1199060 (119909) 119889119909 = 119898 = 0 (6)

Let 119906(119905 119909) and V119898(119905 119909) be the solutions constructed in Theo-rems 1 and 2 respectively Then there exists small 120574 gt 0 suchthat the asymptotics

1003817100381710038171003817119906 (119905) minus V119898 (119905)1003817100381710038171003817Linfin le 119862119905minus12+120574 (7)

are true for 119905 ge 1

Our approach is based on the factorization techniquesDefine the free evolution group U(119905) = Fminus1119890minus(1198941199053)|120585|3F andwrite

U (119905)Fminus1120601 = D119905radic |119905|2120587 intR 119890119894119905(119909120585minus(13)|120585|3)120601 (120585) 119889120585 (8)

where D119905120601 = |119905|minus12120601(119909119905) is the dilation operator Thereis a unique stationary point 120585 = 120583(119909) equiv (119909|119909|)radic|119909| inthe integral int

R119890119894119905(119909120585minus(13)|120585|3)120601(120585)119889120585 which is defined as the

root of the equation 120585|120585| = 119909 for all 119909 isin R Define thescaling operator (B120601)(119909) = 120601(120583(119909)) Hence we find thefollowing decomposition U(119905)Fminus1120601 = D119905B119872V120601 wherethe multiplication factor119872 = 119890(21198941199053)|120578|3 and the deformationoperator

V (119905) 120601 = radic |119905|2120587 intR 119890minus119894119905119878(120585120578)120601 (120585) 119889120585 (9)

where the phase function 119878(120585 120578) = (13)|120585|3 minus (13)|120578|3 minus120578|120578|(120585 minus 120578) Denote A119896 = 119872119896(12119905|120578|)120597120578119872119896 119896 = 0 1 WehaveA1 = A0 + 119894120578 and alsoA1V = V119894120585 [119894120578V] = minusA0Vtherefore we obtain the commutator 120597120578V = minus2119905|120578|[119894120578V]Since 120597120585119878(120585 120578) = 120585|120585| minus 120578|120578| then we get 119894119905[120578|120578|V]120601 =minusV120597120585120601 Also we need the representation for the inverseevolution group FU(minus119905)120601 = Vlowast119872Bminus1Dminus1

119905 where theinverse dilation operator Dminus1

119905 120601 = |119905|12120601(119909119905) the inversescaling operator (Bminus1120601)(120578) = 120601(120578|120578|) and the inversedeformation operator

Vlowast (119905) 120601 = radic2 |119905|120587 int

R

119890119894119905119878(120585120578)120601 (120578) 10038161003816100381610038161205781003816100381610038161003816 119889120578 (10)

We have 119894120585Vlowast120601 = VlowastA1120601 Hence the commutator[119894120585Vlowast] = VlowastA0 Define the new dependent variable 120593 =FU(minus119905)119906(119905) Since FU(minus119905)L = 120597119905FU(minus119905) with L = 120597119905 +(1198943)|120597119909|3 applying the operator FU(minus119905) to (1) substituting119906(119905) = U(119905)Fminus1120593 = D119905B119872V120593 and using the factorizationtechniques we get

120597119905120593 = FU (minus119905)L119906 = 119894120582120585FU (minus119905) (|119906|2 119906)= 119894120582120585Vlowast119872B

minus1D

minus1119905 (1003816100381610038161003816D119905B119872V12059310038161003816100381610038162 D119905B119872V120593)

= 119894120582120585119905minus1Vlowast119872Bminus1 (1003816100381610038161003816B119872V12059310038161003816100381610038162 B119872V120593)

= 119894120582120585119905minus1Vlowast119872(1003816100381610038161003816119872V12059310038161003816100381610038162119872V120593)= 119894120582120585119905minus1Vlowast (1003816100381610038161003816V12060110038161003816100381610038162 V120601)

(11)

since the nonlinearity is gauge invariant Finally we mentionsome important identities The operator J = U(119905)119909U(minus119905) =119909 + 119894119905120597119909|120597119909| plays a crucial role in the large time asymptoticestimates Note thatJ commutes withL that is [JL] = 0To avoid the derivative loss we also use the operator P =3119905120597119905 + 120597119909119909 Note the commutator relation [P 119890minus(1198941199053)|120585|3] = 0

International Journal of Differential Equations 3

with P = 3119905120597119905 minus 120585120597120585 Thus using 119906(119905) = U(119905)Fminus1120593 =D119905119872V120593 = Fminus1119890minus(1198941199053)|120585|3120593 we getP119906 = U(119905)Fminus1P120593 Alsowe have the identityJ = 120597minus1119909 P minus 3119905120597minus1119909 L and [LP] = 3Lholds

2 Estimates in the Uniform Norm

21 Kernels Define the kernel

119860119895 (119905 120578) = radic |119905|2120587 intR 119890minus119894119905119878(120585120578)Θ(120585120578minus1) 120585119895119889120585 (12)

for 120578 = 0 where the cutoff function Θ(119911) isin C2(R) issuch that Θ(119911) = 0 for 119911 le 13 or 119911 ge 3 and Θ(119911) =1 for 23 le 119911 le 32 We change 120585 = 120578119910 then weget 119860119895(119905 120578) = |120578|120578119895radic|119905|2120587 int3

13119890minus119894119905|120578|3119866(119910)Θ(119910)119910119895119889119910 where119878(120578119910 120578) = |120578|3119866(119910) and 119866(119910) = (13)(119910 + 2)(119910 minus 1)2 119910 gt 0

To compute the asymptotics of the kernel 119860119895(119905 120578) for large 119905we apply the stationary phase method (see [39] p 110)

intR

119890119894119911119892(119910)119891 (119910) 119889119910= 119890119894119911119892(1199100)119891 (1199100)radic 2120587119911 100381610038161003816100381611989210158401015840 (1199100)1003816100381610038161003816 119890119894(1205874)sgn11989210158401015840(1199100)

+ 119874 (119911minus32)(13)

for 119911 rarr +infin where the stationary point 1199100 is defined by theequation 1198921015840(1199100) = 0 By virtue of formula (13) with 119892(119910) =minus119866(119910) 119891(119910) = Θ(119910)119910119895 and 1199100 = 1 we get

119860119895 (119905 120578) = 11990512 10038161003816100381610038161205781003816100381610038161003816 120578119895radic2119894 ⟨1199051205783⟩ + 119874(119905121205781+119895 ⟨1199051205783⟩minus1) (14)

for 1199051205783 rarr infin Also we have the estimate |119860119895(119905 120578)| le11986211990512|120578|119895+1⟨1199051205783⟩minus12In the samemanner changing 120578 = 120585119910 we get for the kernel

119860lowast (119905 120585) = radic 2 |119905|120587 intR

119890119894119905119878(120585120578)Θ(120578120585minus1) 10038161003816100381610038161205781003816100381610038161003816 119889120578= 1205852radic2 |119905|120587 int3

13119890119894119905|120585|3119866(119910)Θ(119910) 10038161003816100381610038161199101003816100381610038161003816 119889119910

(15)

for 120585 = 0 where 119878(120585 120585119910) = |120585|3119866(119910) with 119866(119910) = (13)(2119910 +1)(119910 minus 1)2 119910 gt 0Then by virtue of formula (13) with 119892(119910) =119866(119910) 119891(119910) = Θ(119910)|119910| and 1199100 = 1 we obtain119860lowast (119905 120585) = 2119905121205852radic 2119894⟨1199051205853⟩ + 119874(119905121205852 ⟨1199051205853⟩minus1) (16)

for 1199051205853 rarr infin Also we have the estimate |119860lowast(119905 120585)| le119862119905121205852⟨1199051205853⟩minus12

22 Asymptotics for the Operator V In the next lemma weestimate the operator V in the uniform norm Define thecutoff function 1205941(119911) isin C2(R) such that 1205941(119911) = 0 for |119911| ge 3and 1205941(119911) = 1 for |119911| le 2 and 1205942(119911) = 1minus1205941(119911) Consider twooperators

V119895 (119905) 120601 = radic |119905|2120587 intR 119890minus119894119905119878(120585120578)120601 (120585) 120594119895 (120585120578minus1) 119889120585 (17)

so that we haveV(119905)120601 = V1(119905)120601 +V2(119905)120601 for 120578 = 0 Definethe norm 120601Y = 120601Linfin + 119905minus16120597120585120601L2 Lemma 4 The following estimates |V1120585119895120601 minus 119860119895(119905 120578)120601(120578)| le11986211990512|120578|119895+1⟨11990513120578⟩minus74120601Y if 119895 ge 0 and |V2120585119895120601| le11986211990516minus1198953⟨11990513120578⟩119895minus32120601Y if 119895 = 0 1 are valid for all 119905 ge 1120578 = 0Proof We write

V1120585119895120601 minus 119860119895120601 = radic |119905|2120587 intR 119890minus119894119905119878(120585120578) (120601 (120585) minus 120601 (120578))sdot Θ (120585120578minus1) 120585119895119889120585 + radic |119905|2120587 intR 119890minus119894119905119878(120585120578)120601 (120585)sdot (1205941 (120585120578minus1) minus Θ (120585120578minus1)) 120585119895119889120585 = 1198681 + 1198682

(18)

for 120578 = 0 For the summand 1198681 we integrate by parts viaidentity

119890minus119894119905119878(120585120578) = 1198671120597120585 ((120585 minus 120578) 119890minus119894119905119878(120585120578)) (19)

with1198671 = (1 minus 119894119905(120585 minus 120578)120597120585119878(120585 120578))minus1 to get1198681 = 11986211990512 int

R

119890minus119894119905119878(120585120578) (120601 (120585) minus 120601 (120578)) (120585 minus 120578)sdot 120597120585 (1198671Θ(120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R

119890minus119894119905119878(120585120578) (120585 minus 120578)1198671Θ(120585120578minus1)sdot 120585119895120597120585120601 (120585) 119889120585

(20)

We find the estimates100381610038161003816100381610038161198671Θ(120585120578minus1) 12058511989510038161003816100381610038161003816 + 10038161003816100381610038161003816(120585 minus 120578) 120597120585 (1198671Θ(120585120578minus1) 120585119895)10038161003816100381610038161003816le 119862 100381610038161003816100381612057810038161003816100381610038161198951 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2

(21)

in the domain 13 lt 120585120578 lt 3Therefore we obtain100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895sdot int

13lt120585120578lt3

1003816100381610038161003816120601 (120585) minus 120601 (120578)10038161003816100381610038161003816100381610038161003816120585 minus 12057810038161003816100381610038161003816100381610038161003816120585 minus 1205781003816100381610038161003816 1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2

+ 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 int13lt120585120578lt3

1003816100381610038161003816120585 minus 1205781003816100381610038161003816 10038161003816100381610038161003816120597120585120601 (120585)10038161003816100381610038161003816 1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2 (22)


By the Hardy inequality int13lt120585120578lt3

(|120601(120585)minus120601(120578)|2|120585minus120578|2)119889120585 le1198621205971205851206012L2 and by the Cauchy-Schwarz inequality changing120585 = 120578119910 we find

100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int

13lt120585120578lt3

(120585 minus 120578)2 119889120585(1 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2)2)

12

le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895+32 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int3

13

(119910 minus 1)2 119889119910(1 + |119905| 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2)2)

12

le 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 10038161003816100381610038161205781003816100381610038161003816119895+32 ⟨1199051205783⟩minus34

(23)

To estimate the integral 1198682 we integrate by parts via theidentity

119890minus119894119905119878(120585120578) = 1198672120597120585 (120585119890minus119894119905119878(120585120578)) (24)

with1198672 = (1 minus 119894119905120585120597120585119878(120585 120578))minus1 to get1198682 = 11986211990512120601 (0)

sdot intR

119890minus119894119905119878(120585120578)120585120597120585 ((1205941 (120585120578minus1) minus Θ (120585120578minus1))1198672120585119895) 119889120585+ 11986211990512 int

R

119890minus119894119905119878(120585120578)sdot 120601 (120585) minus 120601 (0)120585 1205852120597120585 ((1205941 (120585120578minus1) minus Θ (120585120578minus1))sdot 1198672120585119895) 119889120585 + 11986211990512 int

R

119890minus119894119905119878(120585120578) (1205941 (120585120578minus1)minus Θ (120585120578minus1))1198672120585119895+1120601120585 (120585) 119889120585

(25)

We find the estimates |120585120597120585((1205941(120585120578minus1) minus Θ(120585120578minus1))1198672120585119895)| le119862|120585|119895(1 + 119905|120585|1205782) and10038161003816100381610038161003816(1205941 (120585120578minus1) minus Θ (120585120578minus1))1198672120585119895+110038161003816100381610038161003816+ 100381610038161003816100381610038161205852120597120585 ((1205941 (120585120578minus1) minus Θ (120585120578minus1))1198672120585119895)10038161003816100381610038161003816

le 119862 100381610038161003816100381612058510038161003816100381610038161+1198951 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782 (26)

Then by the Hardy inequality we obtain

100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int|120585|le3|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|le3|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 1199051205851205782)2)12

(27)

We have int|120585|le3|120578|

(|120585|119895119889120585(1 + 119905|120585|1205782)) le 119862|120578|119895+1 int3

0(119910119895119889119910(1+119905|120578|3119910)) le 119862|120578|119895+1⟨1199051205783⟩minus1 log⟨1199051205783⟩ and int

|120585|le3|120578|(|120585|2+2119895119889120585

(1 + 119905|120585|1205782)2) le 119862|120578|2119895+3 int3

0(1199102119895+2119889119910(1 + 119905|120578|3119910)2) le119862|120578|2119895+3⟨1199051205783⟩minus2 Thus we have |1198682| le11986211990512|120601(0)||120578|119895+1⟨1199051205783⟩minus34 +11986211990512120597120585120601L2 |120578|119895+32⟨1199051205783⟩minus1 for allge 1 120578 = 0

To estimateV2120585119895120601 we integrate by parts via identity (24)

V2120585119895120601 = 11986211990512120601 (0)sdot int

R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R

119890minus119894119905119878(120585120578)sdot 120601 (120585) minus 120601 (0)120585 1205852120597120585 (11986721205942 (120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R

119890minus119894119905119878(120585120578)1205851+1198951205942 (120585120578minus1)1198672120601120585 (120585) 119889120585

(28)

We find the estimates |120585120597120585(11986721205942(120585120578minus1)120585119895)| le 119862|120585|119895(1 + 119905|120585|3)and |1205852120597120585(11986721205942(120585120578minus1)120585119895)| + |1205851+1198951205942(120585120578minus1)1198672| le 119862|120585|1+119895(1 +119905|120585|3) in the domain |120585| ge (32)|120578| Then by the Hardyinequality we obtain

10038161003816100381610038161003816V212058511989512060110038161003816100381610038161003816le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int

|120585|ge(32)|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 100381610038161003816100381612058510038161003816100381610038163+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|ge(32)|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 119905 100381610038161003816100381612058510038161003816100381610038163)2)12

(29)

We have int|120585|ge(32)|120578|

(|120585|119895119889120585(1+119905|120585|3)) le 119862|120578|1+119895 intinfin

32(119910119895119889119910(1+119905|120578|31199103)) le 119862119905minus(119895+1)3⟨1199051205783⟩(119895minus2)3 and int

|120585|ge(32)|120578|(|120585|2+2119895119889120585(1 +119905|120585|3)2) le 119862|120578|3+2119895 intinfin

32(1199102+2119895119889119910(1 + 119905|120578|31199103)2) le119862119905minus21198953minus1⟨1199051205783⟩21198953minus1 if 119895 = 0 1 Thus we have |V2120585119895120601| le119862|120601(0)|11990516minus1198953⟨1199051205783⟩(119895minus2)3 + 119862120597120585120601L2119905minus1198953⟨1199051205783⟩1198953minus12 for all119905 ge 1 if 119895 = 0 1 Lemma 13 is proved

23 Asymptotics for the Operator Vlowast We next consider theoperatorVlowast Since Vlowast120601Linfin le 119862|119905|12120578120601L1 and Vlowast120601L2 le119862radic|120578|120601L2 then by the Riesz interpolation theorem (see[40] p 52) we have

1003817100381710038171003817Vlowast1206011003817100381710038171003817L119901 le 119862 |119905|12minus1119901 100381710038171003817100381710038171003817100381610038161003816100381612057810038161003816100381610038161minus1119901 120601100381710038171003817100381710038171003817L119901(119901minus1) (30)

for 2 le 119901 le infin In the next lemma we find the asymptoticsof Vlowast Denote 120585 = 12058511990513 Also define the norm 120601I120572120573 =|120578|120572⟨120578⟩minus120573120597120578120601L2 + |120578|120572minus1⟨120578⟩minus120573120601L2


Lemma 5 Let 14 + 2120573 le 120572 lt 52 minus 2120573 0 le 120573 lt 12Thenthe estimate ⟨120585⟩120573(Vlowast120601 minus 119860lowast120601)Linfin le 119862119905(120572minus1)3120601I120572120573 is validfor all 119905 ge 1Proof We write

Vlowast120601 minus 119860lowast120601= radic2 |119905|120587 intinfin

minusinfin119890119894119905119878(120585120578) (120601 (120578) minus 120601 (120585))Θ (120578120585minus1) 10038161003816100381610038161205781003816100381610038161003816 119889120578

+ radic2 |119905|120587 intinfin

minusinfin119890119894119905119878(120585120578)120601 (120578) (1 minus Θ (120578120585minus1)) 10038161003816100381610038161205781003816100381610038161003816 119889120578

= 1198681 + 1198682

(31)

for 120585 = 0 In the integral 1198681 we use the identity119890119894119905119878(120585120578) = 1198673120597120578 ((120578 minus 120585) 119890119894119905119878(120585120578)) (32)

with1198673 = (1 + 119894119905(120578 minus 120585)120597120578119878(120585 120578))minus1 120597120578119878(120585 120578) = minus2|120578|(120585 minus 120578)and integrate by parts

1198681 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578) 120601 (120578) minus 120601 (120585)120578 minus 120585 10038161003816100381610038161205781003816100381610038161003816120572 ⟨120578⟩minus120573 (120578 minus 120585)2

sdot 120597120578 (1198673100381610038161003816100381612057810038161003816100381610038161minus120572 ⟨120578⟩120573Θ(120578120585minus1)) 119889120578

+ 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578) (120578 minus 120585)1198673

100381610038161003816100381612057810038161003816100381610038161minus120572 ⟨120578⟩120573Θ(120578120585minus1)sdot 120597120578 (10038161003816100381610038161205781003816100381610038161003816120572 ⟨120578⟩minus120573 120601 (120578)) 119889120578

(33)

Then apply the estimates |(120578minus 120585)1198673|120578|1minus120572⟨120578⟩120573Θ(120578120585minus1)| + |(120578minus120585)2⟨120578⟩120573120597120578(1198673|120578|1minus120572Θ(120578120585minus1))| le 119862|120585|1minus120572⟨120585⟩120573|120578minus120585|(1+119905|120585|(120578minus120585)2) in the domain 13 le 120578120585 le 3 If |120581(119909)| le 119862|120581(119910119909)| for all119910 isin (0 1) then we find the Hardy inequality

10038171003817100381710038171003817100381710038171003817120581 (119909)119909 (120601 (119909) minus 120601 (0))10038171003817100381710038171003817100381710038171003817L2 =10038171003817100381710038171003817100381710038171003817120581 (119909)119909 int119909

01206011015840 (119910) 11988911990511991010038171003817100381710038171003817100381710038171003817L2

= 100381710038171003817100381710038171003817100381710038171003817120581 (119909) int1

01206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le 119862 100381710038171003817100381710038171003817100381710038171003817int1

0120581 (119909119910) 1206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le int1

0

10038171003817100381710038171003817120581 (119909119910) 1206011015840 (119909119910)10038171003817100381710038171003817L2 119889119905 le int1

0119910minus12119889119910 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

le 2 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

(34)

Hence int13le120578120585le3

(|120601(120578) minus 120601(120585)|2(120578 minus 120585)2)|120578|2120572⟨120578⟩minus2120573119889120578 le1198621206012I120572120573 Therefore changing 120578 = 119910120585 we have⟨120585⟩120573 100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 100381610038161003816100381612058510038161003816100381610038161minus120572 ⟨120585⟩2120573 10038171003817100381710038171206011003817100381710038171003817I120572120573sdot (int

13le120578120585le3

(120578 minus 120585)2 119889120578(1 + 119905 10038161003816100381610038161205851003816100381610038161003816 (120578 minus 120585)2)2)

12

le 11986211990512 100381610038161003816100381612058510038161003816100381610038161minus120572 ⟨120585⟩2120573 10038171003817100381710038171206011003817100381710038171003817I120572120573sdot (int3

13

100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2 119889119910(1 + 119905 100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2)2)12

le 11986211990512 1003816100381610038161003816120585100381610038161003816100381652minus120572

sdot ⟨120585⟩2120573minus94 10038171003817100381710038171206011003817100381710038171003817I120572120573 le 119862119905(120572minus1)3 10038171003817100381710038171206011003817100381710038171003817I120572120573

(35)

if 14 + 2120573 le 120572 le 52 In the integral 1198682 using the identity119890119894119905119878(120585120578) = 1198674120597120578 (120578119890119894119905119878(120585120578)) (36)

with 1198674 = (1 + 119894119905120578120597120578119878(120585 120578))minus1 120597120578119878(120585 120578) = minus2|120578|(120585 minus 120578) weintegrate by parts

1198682 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)

sdot 10038161003816100381610038161205781003816100381610038161003816120572 120601 (120578)120578 ⟨120578⟩120573 1205782120597120578 (1198674 (1 minus Θ (120578120585minus1)) 100381610038161003816100381612057810038161003816100381610038161minus120572sdot ⟨120578⟩120573) 119889120578 + 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)1205781198674 (1

minus Θ (120578120585minus1)) 100381610038161003816100381612057810038161003816100381610038161minus120572 ⟨120578⟩120573 120597120578 (10038161003816100381610038161205781003816100381610038161003816120572 ⟨120578⟩minus120573 120601 (120578)) 119889120578

(37)

Then using the estimates 120597120578119878(120585 120578) = 119874(|120578|(120585 + |120578|)) in thedomains 120578120585 lt 13 and 120578120585 ge 3 we get1003816100381610038161003816100381610038161205782120597120578 (1198674 (1 minus Θ (120578120585minus1)) 100381610038161003816100381612057810038161003816100381610038161minus120572 ⟨120578⟩120573)100381610038161003816100381610038161003816

+ 1003816100381610038161003816100381610038161205781198674 (1 minus Θ (120578120585minus1)) 100381610038161003816100381612057810038161003816100381610038161minus120572 ⟨120578⟩120573100381610038161003816100381610038161003816le 119862 100381610038161003816100381612057810038161003816100381610038162minus120572 ⟨120578⟩1205731 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)

(38)

Therefore by the Hardy inequality

⟨120585⟩120573 100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 ⟨120585⟩120573 100381710038171003817100381710038171003817120597120578 (10038161003816100381610038161205781003816100381610038161003816120572 ⟨120578⟩minus120573 120601 (120578))100381710038171003817100381710038171003817L2sdot (intinfin

minusinfin

⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2)12 le 11986211990512 10038171003817100381710038171206011003817100381710038171003817I120572120573

sdot (intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2 )12

(39)


We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1

01205784minus4120573minus2120572 ⟨120578⟩6120573minus6 119889120578

+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513

01205784minus4120573minus2120572 ⟨120578⟩6120573minus6 119889120578 + 11986211990541205733minus2

le 119862119905(2120572minus5)3

(40)

if 14 + 2120573 le 120572 lt 52 minus 2120573 0 le 120573 lt 12 Therefore we get⟨120585⟩120573|1198682| le 119862119905(120572minus1)3120601I120572120573 Lemma 5 is proved

3 Commutators withV

First we estimate the Fourier type integral

W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)

in the L2-norm In the particular factorized case 119902(119905 120585 120578) =1199021(120585)1199022(120578) with estimate |1199022(120583(119909))| le |120583(119909)|12 we find1003817100381710038171003817W1206011003817100381710038171003817L2 le 11990512 10038171003817100381710038171003817100381710038171198721199022 (120578)sdot int

R

1198901198941199051205782120585119890minus(1198941199053)|120585|31199021 (120585) 120601 (120585) 1198891205851003817100381710038171003817100381710038171003817L2le 11990512 10038171003817100381710038171003817100381710038171199022 (120583 (119909)) 1003816100381610038161003816120583 (119909)1003816100381610038161003816minus12sdot int

R

119890119894119905119909120585119890minus(1198941199053)|120585|31199021 (120585) 120601 (120585) 1198891205851003817100381710038171003817100381710038171003817L2le 119862 100381710038171003817100381710038171003817119890minus(1198941199053)|120585|31199021120601100381710038171003817100381710038171003817L2 = 119862 100381710038171003817100381711990211206011003817100381710038171003817L2

(42)

Next we obtain a more general result

Lemma 6 Suppose that |(120578120597120578)119896119902(119905 120585 120578)| le 119862120578]⟨120578⟩minus] for all120585 120578 isin R 119896 = 0 1 2 where ] isin (0 1) Then the estimate|120578|12W120601L2 le 119862120601L2 is true for all 119905 ge 1

Proof We write1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 W1206011003817100381710038171003817100381710038172L2 = 119862119905intR 119889120578 10038161003816100381610038161205781003816100381610038161003816 intR 119889120585sdot int

R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)

where the kernel119870(119905 120585 120577) = intR119889120578|120578|119890119894119905120578|120578|(120585minus120577)119902(119905 120585 120578)119902(119905 120577 120578)

Integrating two times by parts via the identity 119890119894119905120578|120578|(120585minus120577) =119867120597120578(120578119890119894119905120578|120578|(120585minus120577)) with119867 = (1 + 2119894119905120578|120578|(120585 minus 120577))minus1 we get119870 (119905 120585 120577) = int

R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)

Since |120578120597120578(119867120578120597120578(119867|120578|119902(119905 120585 120578)119902(119905 120577 120578)))| le 119862|120578|1205782]⟨120578⟩minus2](1 + 1199051205782|120585 minus 120577|)2 we get1003816100381610038161003816119870 (119905 120585 120577)1003816100381610038161003816le 119862int1

0

1205782] ⟨120578⟩minus2] 10038161003816100381610038161205781003816100381610038161003816 119889120578(1 + 1199051205782 1003816100381610038161003816120585 minus 1205771003816100381610038161003816)2+ 119862intinfin

1

⟨120578⟩minus2] 10038161003816100381610038161205781003816100381610038161003816 119889120578(1 + 1199051205782 1003816100381610038161003816120585 minus 1205771003816100381610038161003816)2le 119862119905minus23 int1

0

1205781+2]119889120578(1 + 120578211990513 1003816100381610038161003816120585 minus 1205771003816100381610038161003816)2+ 119862119905minus23 intinfin

1

1205781minus2]119889120578(1 + 119905131205782 1003816100381610038161003816120585 minus 1205771003816100381610038161003816)2+ 119862119905minus2]3 (1003816100381610038161003816120585 minus 1205771003816100381610038161003816 119905)]minus1 intinfin

11990512|120585minus120577|121205781minus2] ⟨120578⟩minus4 119889120578

le 119862119905minus23 ⟨(120585 minus 120577) 11990513⟩minus1minus]

+ 119862119905minus23 (1003816100381610038161003816120585 minus 1205771003816100381610038161003816 11990513)]minus1 ⟨(120585 minus 120577) 11990513⟩minus1

+ 119862119905minus2]3 (1003816100381610038161003816120585 minus 1205771003816100381610038161003816 119905)]minus1 ⟨(120585 minus 120577) 119905⟩minus1minus]

(45)

Thenby theCauchy-Schwarz inequality andYoung inequalitywe obtain1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 W1206011003817100381710038171003817100381710038172L2 le 11986211990513 10038171003817100381710038171206011003817100381710038171003817L2

sdot 1003817100381710038171003817100381710038171003817int ⟨(120585 minus 120577) 11990513⟩minus1minus] 1003816100381610038161003816120601 (120577)1003816100381610038161003816 1198891205771003817100381710038171003817100381710038171003817L2 + 11986211990513 10038171003817100381710038171206011003817100381710038171003817L2sdot 1003817100381710038171003817100381710038171003817int (1003816100381610038161003816120585 minus 1205771003816100381610038161003816 11990513)]minus1 ⟨(120585 minus 120577) 11990513⟩minus1 1003816100381610038161003816120601 (120577)1003816100381610038161003816 1198891205771003817100381710038171003817100381710038171003817L2+ 1198621199051minus2]3 10038171003817100381710038171206011003817100381710038171003817L2


sdot 1003817100381710038171003817100381710038171003817int (1003816100381610038161003816120585 minus 1205771003816100381610038161003816 119905)]minus1 ⟨(120585 minus 120577) 119905⟩minus1minus] 1003816100381610038161003816120601 (120577)1003816100381610038161003816 1198891205771003817100381710038171003817100381710038171003817L2le 11986211990513 100381710038171003817100381710038171003817⟨12058511990513⟩minus1minus]100381710038171003817100381710038171003817L1 100381710038171003817100381712060110038171003817100381710038172L2+ 11986211990513 100381710038171003817100381710038171003817(10038161003816100381610038161205851003816100381610038161003816 11990513)]minus1 ⟨12058511990513⟩minus1100381710038171003817100381710038171003817L1 100381710038171003817100381712060110038171003817100381710038172L2+ 1198621199051minus2]3 10038171003817100381710038171003817(10038161003816100381610038161205851003816100381610038161003816 119905)]minus1 ⟨120585119905⟩minus1minus]10038171003817100381710038171003817L1 100381710038171003817100381712060110038171003817100381710038172L2 le 119862 100381710038171003817100381712060110038171003817100381710038172L2

(46)

if ] isin (0 1) Lemma 14 is proved

Next we estimateW120601(0)Lemma 7 Suppose that |(120585120597120585)119896119902(119905 120585 120578)| le 119862(|120578| + |120585|)minus1 forall 120585 120578 isin R 119896 = 0 1Then the estimate |120578|12W1L2 le 11986211990516is true

Proof As in the proof of Lemma 13 we decompose

W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)sdot (1205941 (120585120578minus1) minus Θ (120585120578minus1)) 119889120585+ 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)

for 120578 = 0 In the first summand 1198681 we integrate by parts viaidentity (19) to get

1198681 = 11986211990512 intR

119890minus119894119905119878(120585120578) (120585 minus 120578)sdot 120597120585 (1198671119902 (119905 120585 120578)Θ (120585120578minus1)) 119889120585 (48)

Using the estimate |(120585 minus 120578)120597120585(1198671119902(119905 120585 120578)Θ(120585120578minus1))| le119862|120578|minus1(1 + 119905|120578|(120585 minus 120578)2) then changing 120585 = 120578119910 we obtain100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int

13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)

Thus we get |120578|121198681L2 le 11986211990512|120578|12⟨120578⟩minus32L2 le 11986211990516 Toestimate the integral 1198682 we integrate by parts via identity (24)to get

1198682 = 11986211990512 intR

119890minus119894119905119878(120585120578)120585120597120585 ((1205941 (120585120578minus1) minus Θ (120585120578minus1))sdot 1198672119902 (119905 120585 120578)) 119889120585

(50)

Wefind the estimates |120585120597120585((1205941(120585120578minus1)minusΘ(120585120578minus1))1198672119902(119905 120585 120578))| le119862|120578|minus1(1 + 119905|120585|1205782) then we obtain

100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)

Thus as above we get |120578|121198682L2 le 11986211990516 for all 119905 ge 1 if ] gtminus12 To estimate 1198683 we integrate by parts via identity (24)1198683 = 11986211990512 int

R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)

We find the estimates |120585120597120585(11986721205942(120585120578minus1)119902(119905 120585 120578))| le 119862(|120578| +|120585|)(1+119905|120585|3) in the domain |120585| ge (32)|120578| and then we obtain100381610038161003816100381611986831003816100381610038161003816 le 11986211990512 intinfin

(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32

119889119910(1 + 119910) (1 + 119905 100381610038161003816100381612057810038161003816100381610038163 1199103)+ 119862119905minus12 ⟨120578⟩minus1 intinfin

(32)⟨120578⟩120585minus3119889120585

le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus120574 ⟨120578⟩120574minus3 + 119862119905minus12 ⟨120578⟩minus3

(53)

if 0 lt 120574 lt 1Thus we find

1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 1198683100381710038171003817100381710038171003817L2 le 11986211990512 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 10038161003816100381610038161205781003816100381610038161003816minus120574 ⟨120578⟩120574minus3100381710038171003817100381710038171003817L2+ 119862119905minus12 100381710038171003817100381710038171003817⟨120578⟩minus52100381710038171003817100381710038171003817L2 le 11986211990516

(54)

for all 119905 ge 1 Lemma 7 is proved

In the next lemma we estimate the commutator [120578V1]Define the norm 120601Y = 120601Linfin + 119905minus16120597120585120601L2 Lemma 8 Let 119895 ge 0 ] isin (0 1) Then the estimate|120578|32minus119895120578]⟨120578⟩minus]119905[120578V1]120585119895120601L2 le 11986211990516120601Y is true for all119905 ge 1Proof For 120578 = 0 we integrate by parts

100381610038161003816100381612057810038161003816100381610038161minus119895 120578] ⟨120578⟩minus] 119905 [120578V1] 120585119895120601 = 100381610038161003816100381612057810038161003816100381610038161minus119895 120578] ⟨120578⟩minus]sdot radic |119905|2120587 intR 119890minusi119905119878(120585120578)120597120585 ( 120578 minus 120585119894120597120585119878 (120585 120578)1205941 (120585120578minus1)sdot 120585119895120601 (120585)) 119889120585 = radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578)


sdot 120597120585120601 (120585) 119889120585 + radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199022 (119905 120585 120578)sdot 120601 (120585) minus 120601 (0)120585 119889120585 + 120601 (0)sdot radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

(55)

where 1199021(119905 120585 120578) = 120578]⟨120578⟩minus]|120578|1minus119895((120578 minus 120585)119894120597120585119878(120585120578))1205851198951205941(120585120578minus1) 1199022(119905 120585 120578) = 1205851205971205851199021(119905 120585 120578) and 1199023(119905 120585 120578) =1205971205851199021(119905 120585 120578) Since 120597120585119878(120585 120578) = 120585|120585| minus 120578|120578| and (120578 minus 120585)120597120585119878(120585 120578) = (|120585| + |120578|)(1205852 + 1205782 + |120585||120578| + 120578120585) we have1199021 (119905 120585 120578) = 100381610038161003816100381612057810038161003816100381610038161minus119895 120578] ⟨120578⟩minus] 120578 minus 120585120585 10038161003816100381610038161205851003816100381610038161003816 minus 120578 10038161003816100381610038161205781003816100381610038161003816 1205851198951205941 (120585120578minus1)

= 119874 (120578] ⟨120578⟩minus]) (56)

and similarly 1199022(119905 120585 120578) = 1205851205971205851199021(120585 120578) = 119874(120578]⟨120578⟩minus]) forall 120585 120578 isin R in the domain |120585| le 3|120578| Hence we have|(120578120597120578)119896119902119897(119905 120585 120578)| le 119862120578]⟨120578⟩minus] for all 120585 120578 isin R 119896 = 0 1 2and 119897 = 1 2 where ] isin (0 1) and by Lemma 14

100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)

and by the Hardy inequality and Lemma 14

100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199022 (119905 120585 120578) 120601 (120585) minus 120601 (0)120585 11988912058510038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 10038171003817100381710038171003817100381710038171003817120601 (120585) minus 120601 (0)120585

10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have

1199023 (119905 120585 120578)= 100381610038161003816100381612057810038161003816100381610038161minus119895 120578] ⟨120578⟩minus] 120597120585 ( 120578 minus 120585(120585 10038161003816100381610038161205851003816100381610038161003816 minus 120578 10038161003816100381610038161205781003816100381610038161003816) 1205851198951205941 (120585120578minus1))= 119874((10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816)minus1 ⟨120578⟩120573+119895minus1)

(59)

for all 120585 120578 isin R in the domain |120585| le 3|120578| Hencewe get |(120585120597120585)1198961199023(119905 120585 120578)| le 119862(|120578| + |120585|)minus1 for all 120585 120578 isinR 119896 = 0 1 Therefore applying Lemma 7 we obtain|120578|12radic|119905|2120587 int

R119890minus119894119905119878(120585120578)1199023(119905 120585 120578)119889120585L2 le 11986216 Lemma 8 is

proved

In the next lemma we estimate the operatorV2Lemma 9 Let 119895 = 0 1 2 ] isin (0 1) Then the estimate|120578|52minus119895120578]⟨120578⟩minus]119905V2120585119895120601L2 le 11986211990516120601Y is true for all 119905 ge 1

Proof We integrate by parts

120578] ⟨120578⟩minus] 100381610038161003816100381612057810038161003816100381610038162minus119895 119905V2120585119895120601 = 120578] ⟨120578⟩minus] 100381610038161003816100381612057810038161003816100381610038162minus119895sdot radic |119905|2120587 intR 119890minus119894119905119878(120585120578)120597120585(120585

1198951205942 (120585120578minus1)119894120597120585119878 (120585 120578) 120601 (120585))119889120585= radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585+ radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199022 (119905 120585 120578) 120601 (120585) minus 120601 (0)120585 119889120585+ 120601 (0)radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

(60)

where 1199021(119905 120585 120578) = (120578]⟨120578⟩minus]|120578|2minus119895120585119895119894120597120585119878(120585 120578))1205942(120585120578minus1)1199022(119905 120585 120578) = 1205851205971205851199021(119905 120585 120578) and 1199023(119905 120585 120578) = 1205971205851199021(119905 120585 120578) Wefind

1199021 (119905 120585 120578) = 119874 (120578] ⟨120578⟩minus] 100381610038161003816100381612057810038161003816100381610038162minus119895 10038161003816100381610038161205851003816100381610038161003816119895minus2)= 119874 (120578] ⟨120578⟩minus]) (61)

and 1199022(119905 120585 120578) = 119874(120578]⟨120578⟩minus]) for all 120585 120578 isin R in thedomain |120585| ge (32)|120578| if 119895 = 0 1 2Then |(120578120597120578)119896119902119897(119905 120585 120578)| le119862120578]⟨120578⟩minus] for all 120585 120578 isin R 119896 = 0 1 2 and 119897 = 1 2 Hence byLemma 14 we find10038171003817100381710038171003817100381710038171003817100381710038171003817

1003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 11988912058510038171003817100381710038171003817100381710038171003817100381710038171003817L2

le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)

and by the Hardy inequality


10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)

Alsowe have 1199023(119905 120585 120578) = 119874((|120578|+|120585|)minus1) for all 120585 120578 isin R in thedomain |120585| ge (32)|120578| if 119895 = 0 1 2Hence |(120585120597120585)1198961199023(119905 120585 120578)| le119862(|120578| + |120585|)minus1 for all 120585 120578 isin R 119896 = 0 1 and by Lemma 7

100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved

In the next lemma we estimate the derivative 120597120578VLemma 10 Let 119895 = 0 1 ] isin (0 1) Then the estimate|120578|12minus119895120578]⟨120578⟩minus]120597120578V120585119895120601L2 le 11986211990516120601Y is true for all 119905 ge 1


Proof Since A1V = V119894120585 with A1 = A0 + 119894120578A0 = (12119905|120578|)120597120578 then we obtain the commutator 120597120578V =2119905|120578|[119894120578V] AlsoV = V1 +V2Hence1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612minus119895 120578] ⟨120578⟩minus] 120597120578V120585119895120601100381710038171003817100381710038171003817L2

le 119862119905 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381632minus119895 120578] ⟨120578⟩minus] [120578V1] 120585119895120601100381710038171003817100381710038171003817L2+ 119862119905 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381632minus119895 120578] ⟨120578⟩minus] V2120585119895120601100381710038171003817100381710038171003817L2+ 119862119905 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612minus119895 120578] ⟨120578⟩minus] V2120585119895+1120601100381710038171003817100381710038171003817L2

(65)

By Lemma 8 we find |120578|32minus119895120578]⟨120578⟩minus]119905[120578V1]120585119895120601L2 le11986211990516120601Y for all 119905 ge 1 if 119895 ge 0 ] isin (0 1)Also by Lemma 9 we get |120578|52minus119895120578]⟨120578⟩minus]119905V2120585119895120601L2 +|120578|32minus119895120578]⟨120578⟩minus]119905V2120585119895+1120601L2 le 11986211990516120601Y for all 119905 ge 1 if119895 = 0 1 ] isin (0 1) Hence the result of the lemma followsLemma 10 is proved

4 A Priori Estimates

Local existence and uniqueness of solutions to Cauchyproblem (1) were shown in [19 20] when 1199060 isin H1 By usingthe local existence result we have the following

Theorem 11 Assume that the initial data 1199060 isin H1 cap H01Then there exists a unique local solution 119906 of Cauchy problem(1) such thatU(minus119905)119906 isin C([0 119879]H1 capH01)

We can take 119879 gt 1 if the data are small in H1 cap H01 andwe may assume that 119906X1 le 120576 To get the desired results weprove a priori estimates of solutions uniformly in timeDefinethe following norm

119906X119879 = sup119905isin[1119879]

(1003817100381710038171003817120593 (119905)1003817100381710038171003817Linfin + 119905minus16 J119906 (119905)L2+ 119905(13)(1minus1119901) 119906 (119905)L119901)

(66)

whereJ = U(119905)119909U(minus119905) 120593(119905) = FU(minus119905)119906(119905) and 119901 gt 4First we obtain the large time asymptotic behavior of the

nonlinearityFU(minus119905)120597119909(|119906|2119906)Lemma 12 The asymptotics 119905FU(minus119905)120597119909(|119906|2119906) =119894120585|120585|5⟨120585⟩minus6|120593|2120593 + 119874(120585⟨120585⟩minus]1205933Y) is true for all 119905 ge 1and 120585 isin R where 120593(119905) = FU(minus119905)119906(119905) and ] gt 0 is small

Proof In view of factorization formula (11) we find119905FU(minus119905)120597119909(|119906|2119906) = 119894120585Vlowast(119905)|120595|2120595 = Vlowast(119905)A1(119905)|120595|2120595where 120595 = V120593Then by Lemma 5 with 120572 = 12 + ] 120573 = 2]and ] gt 0 small we get

119905FU (minus119905) 120597119909 (|119906|2 119906) = 119894120585Vlowast (119905) 100381610038161003816100381612059510038161003816100381610038162 120595 = 119894120585119860lowast 100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905minus12+]3120585 (1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612+] ⟨120578⟩minus2] 120597120578 (100381610038161003816100381612059510038161003816100381610038162 120595)100381710038171003817100381710038171003817L2+ 10038171003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816minus12+] ⟨120578⟩minus2] 100381610038161003816100381612059510038161003816100381610038162 120595100381710038171003817100381710038171003817L2))

(67)

in the case of |120585| lt 119905minus13 and119905FU (minus119905) 120597119909 (|119906|2 119906) = V

lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]

sdot (1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612+] ⟨120578⟩minus2] 120597120578A1 (100381610038161003816100381612059510038161003816100381610038162 120595)100381710038171003817100381710038171003817L2+ 10038171003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816minus12+] ⟨120578⟩minus2] A1 (100381610038161003816100381612059510038161003816100381610038162 120595)100381710038171003817100381710038171003817L2))

(68)

in the case of |120585| gt 119905minus13 Via identity |120578|] = 119905minus]3120578]⟨120578⟩] weconsider the remainder terms1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612+] ⟨120578⟩minus2] 120597120578 (100381610038161003816100381612059510038161003816100381610038162 120595)100381710038171003817100381710038171003817L2

le 119862 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612+] ⟨120578⟩minus2] 100381610038161003816100381612059510038161003816100381610038162 120597120578120595100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 10038161003816100381610038161205951003816100381610038161003816210038171003817100381710038171003817Linfinsdot 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 120578] ⟨120578⟩minus] 120597120578120595100381710038171003817100381710038171003817L2 le 11986211990516minus]3 10038171003817100381710038171206011003817100381710038171003817Ysdot 1003817100381710038171003817100381710038161003816100381610038161205951003816100381610038161003816210038171003817100381710038171003817Linfin 10038171003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816minus12+] ⟨120578⟩minus2] 120597120578A1 (100381610038161003816100381612059510038161003816100381610038162 120595)100381710038171003817100381710038171003817L2le 119862 10038171003817100381710038171003817100381610038161003816100381612057810038161003816100381610038161+] 120578minus] ⟨120578⟩minus] 10038161003816100381610038161205951003816100381610038161003816210038171003817100381710038171003817Linfinsdot 10038171003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816minus12 120578] ⟨120578⟩minus] 1205971205781205951

100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951

10038171003817100381710038171003817Linfinsdot 1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 120578] ⟨120578⟩minus] 120597120578120595100381710038171003817100381710038171003817L2 le 11986211990516minus]3 10038171003817100381710038171206011003817100381710038171003817Ysdot (10038171003817100381710038171003817120578 10038161003816100381610038161205951003816100381610038161003816210038171003817100381710038171003817Linfin + 10038171003817100381710038171205951205951

1003817100381710038171003817Linfin)

(69)

where 1205951 = V119894120585120593 By Lemma 10 with 119895 = 0 1 we have1003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612 120578] ⟨120578⟩minus] 120597120578120595100381710038171003817100381710038171003817L2 + 10038171003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816minus12 120578] ⟨120578⟩minus] 1205971205781205951

100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)

Using Lemma 13 we get |120595| le 119862(11990512|120578|⟨120578⟩minus32 +11990516⟨120578⟩minus32)120601Y le 11986211990516⟨120578⟩minus12120593Y and |1205951| le119862(11990512|120578|2⟨120578⟩minus32 + 119905minus16⟨120578⟩minus12)120601Y le 119862(|120578|12 + 119905minus16)120593YHence 1205952Linfin le 119862119905131205932Y and 120578|120595|2Linfin + 1205951205951Linfin le1198621205932Y Also we find |120578|minus12+]⟨120578⟩minus2]|120595|2120595L2 le119862119905121205933Y|120578|minus12+]⟨120578⟩minus2]minus32L2 le 11986211990512minus]31205933Y and|120578|minus12+]⟨120578⟩minus2]A1(|120595|2120595)L2 le 119862119905161205933Y|120578|minus12+]⟨120578⟩minus2]minus12L2 le11986211990516minus]31205933Y Therefore we obtain 119905FU(minus119905)120597119909(|119906|2119906) =119894120585119860lowast|120595|2120595 + 119874(120585⟨120585⟩minus2]1205933Y) for |120585| lt 119905minus13 and119905FU(minus119905)120597119909(|119906|2119906) = 119860lowastA1|120595|2120595 + 119874(120585⟨120585⟩minus2]1205933Y)for |120585| gt 119905minus13 Next by Lemma 13 we have 120595119895(119905 120585) =(11990512|120585|(119894120585)119895radic2119894⟨1199051205853⟩)120593(120585) + 119874(11990516minus1198953⟨120585⟩119895minus34120593Y) for119895 = 0 1 Then we get 119894120585119860lowast|120595|2120595 = 119894120585|120585|5⟨120585⟩minus6|120593|2120593 +


119874(120585⟨120585⟩minus]1205933Y) and 119860lowastA1|120595|2120595 = 119894120585|120585|5⟨120585⟩minus6|120593|2120593 +119874(120585⟨120585⟩minus]1205933Y) Lemma 12 is proved

Next we estimate the solutions in the norm X119879Lemma 13 Assume that 119906X1 le 120576 holds Then there exists120576 gt 0 such that the estimate 119906X119879 lt 119862120576 is true for all 119879 gt 1Proof By continuity of the norm 119906X119879 with respect to 119879arguing by the contradiction we can find the first time 119879 gt 0such that 119906X119879 = 119862120576 To prove the estimate for the normsup119905isin[1119879]120593Linfin lt 119862120576 we use (11) Then in view of Lemma 12we get

120597119905120593 = 120582FU (minus119905) 120597119909 (|119906|2 119906)= 119894120582119905minus1120585 10038161003816100381610038161003816120585100381610038161003816100381610038165 ⟨120585⟩minus6 100381610038161003816100381612059310038161003816100381610038162 120593 + 119874(1205763119905minus1 120585 ⟨120585⟩minus]) (71)

For the case of |120585| lt 119905minus13 we can integrate |120593(119905 120585)| le |120593(1 120585)|+119862|120585|1205933Y int119905

1120591minus23119889120591 le 120576 + 119862120585119905131205763 le 120576 + 1198621205763 For the

case of |120585| ge 119905minus13 multiplying by 120593 and taking the real partof the result we obtain 120597119905(|120593(119905 120585)|2) = 119874(119905minus1120585⟨120585⟩minus]1205934Y)Integrating in time we obtain

1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)

Therefore FU(minus119905)119906(119905)Linfin lt 119862120576 Applying estimate ofLemma 4 we find |120597119895119909119906(119905 119909)| le 119862120576⟨119909119905minus13⟩1198952minus14119905minus13minus1198953|119906(119905 119909)120597119909119906(119905 119909)| le 1198621205762119905minus1 and

119906L119901 le 119862119905minus13 119906X119879 1003817100381710038171003817100381710038171003817⟨119909119905minus13⟩minus141003817100381710038171003817100381710038171003817L119901le 119862120576119905minus(13)(1minus1119901) (73)

if 119901 gt 4 Consider a priori estimates for J119906(119905)L2 Usingthe identity 120597minus1119909 P119906 minus J119906 = 3119905120597minus1119909 L119906 we get J119906L2 le119862120597minus1119909 P119906L2 +1198621199051199063L6 le 119862120597minus1119909 P119906L2 +119862120576311990516Applying theoperator 120597minus1119909 P to (1) in view of the commutators [LP] =3L [P 120597119909] = minus120597119909 we get L120597minus1119909 P119906 = 120597minus1119909 (P + 3)L119906 =120582120597minus1119909 (P + 3)120597119909(|119906|2119906) = 120582(P + 2)(|119906|2119906)Then by the energymethod we obtain

119889119889119905 10038171003817100381710038171003817120597minus1119909 P119906100381710038171003817100381710038172L2 le 119862 10038171003817100381710038171199061199061199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119906100381710038171003817100381710038172L2

+ 1199063L6 10038171003817100381710038171003817120597minus1119909 P11990610038171003817100381710038171003817L2le 1198621205762119905minus1 10038171003817100381710038171003817120597minus1119909 P119906100381710038171003817100381710038172L2+ 1198621205763119905minus56 10038171003817100381710038171003817120597minus1119909 P11990610038171003817100381710038171003817L2

(74)

from which it follows that 120597minus1119909 P119906L2 le 1205761199051198621205762 + 119862120576311990516Therefore J119906L2 le 119862120597minus1119909 P119906L2 + 1198621199051199063L6 le 119862120576311990516 forall 119905 isin [1 119879] Thus we obtain 119906X119879 lt 119862120576 Lemma 13 isproved

5 Proof of Theorem 1

By Lemma 13 we see that a priori estimate 119906X119879 le 119862120576 istrue for all 119879 gt 0 Therefore the global existence of solutionsof Cauchy problem (1) satisfying the estimate 119906Xinfin le119862120576 follows by a standard continuation argument by localexistence Theorem 11


In this section we prove the existence of a unique self-similar solution V119898(119905 119909) equiv 119905minus13119891119898(119909119905minus13) for (1) whichis uniquely determined by the total mass condition 119898 =(1radic2120587) intR V119898(119905 119909)119889119909 = 0 Define the operators

V120601 = radic 12120587 intR 119890minus119894119878(120585120578)120601 (120585) 119889120585V

lowast120601 = radic 2120587 intR 119890119894119878(120585120578)120601 (120578) 10038161003816100381610038161205781003816100381610038161003816 119889120578(75)

Then for the self-similar solutions V119898(119905 119909) =119905minus13119891119898(119909119905minus13) = D119905B119872V120593119898 where 120593119898(119905 120585) equivFU(minus119905)V119898(119905) we find that 120593119898 have a self-similar formthat is 120593119898(119905 120585) equiv 120601119898(120585) with 120585 = 12058511990513 Using the relation120597119905120593119898(119905 120585) = (13)119905minus11205781206011015840

119898(120578) by factorization formula (11)we get (13)120578120597120578120601119898(120578) = 119894120582120578Vlowast|V120601119898|2V120601119898 Therefore120597120578120601119898(120578) = 3119894120582Vlowast|V120601119898|2V120601119898 equiv 119865(120601119898) Note that 119865(120601119898)is not in L2 Therefore we need the approximate equationDefine Θ(120578) = 1 for |120578| le 1 and Θ(120578) = 0 for |120578| gt 2and denote Θ119877(120578) = Θ(120578119877) Also define the approximateequation

120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)

Let us show a priori estimate 120601119898119877Z = 120601119898119877Linfin +120597120578120601119898119877L2 le 3|119898| uniformly in 119877 Applying Lemma 12 with119905 = 1 we get120597120578120601119898119877 (120578) = 3119894120582 100381610038161003816100381612057810038161003816100381610038165 ⟨120578⟩minus6 1003816100381610038161003816Θ119877120601119898119877 (120578)10038161003816100381610038162Θ119877120601119898119877 (120578)

+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)

Integrating with respect to 120578 we obtain 120597120578120601119898119877L2 le1198621206011198981198773Z Also multiplying by 120601119898119877 and integrating withrespect to 120578 we get |120601119898119877(120578)| = 119898 + 119874(int120578

0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)

Hence 120601119898119877Linfin le |119898| +1198621206011198981198773Z Thus we obtain 120601119898119877Z le2|119898| + 1198621206011198981198773Z le 3|119898| for some small 119898 Taking thelimit 119877 rarr infin we find that there exists a unique solution


120601119898 of equation 120597120578120601119898(120578) = 3119894120582Vlowast|V120601119898|2V120601119898 in Z By thedefinition of 120601119898(120578) we obtain 120597120585120593119898L2 = 11990516120597120578120601119898L2 le119862|119898|11990516 and 120593119898Linfin le 3|119898| In the same way as in the proofof (73) we have L119901 estimate of V119898 for 119901 gt 47 Proof of Theorem 3

Define the norm 119906Y119879 = sup119905isin[1119879](11990512minus120574119906Linfin + 119905minus120574J119906L2)with a small 120574 gt 0Lemma 14 Suppose that 119906119895X119879 le 119862120576 119895 = 1 2 where 120576 issufficiently small Let 1205931(119905 0) = 1205932(119905 0) for 119895 = 1 2 119905 ge 1where 120593119895(119905 120585) = FU(minus119905)119906119895(119905) Let 1199062 = 119905minus13119891(119909119905minus13) be aself-similar solution Then the estimate 1199061 minus 1199062Y119879 lt 119862120576 istrue for all 119879 gt 1Proof By the continuity of the norm 1199061 minus1199062Y119879 with respectto 119879 arguing by the contradiction we can find for the firsttime119879 gt 0 such that 1199061minus1199062Y119879 = 119862120576Wedenote119908 = 1205931minus1205932

and 119910 = 1199061 minus 1199062 Applying estimate of Lemma 4 we find

1003816100381610038161003816119910 (119905 119909)1003816100381610038161003816 le 119862119905minus13 ⟨11990912119905minus16⟩minus12 10038161003816100381610038161003816119908 (119905 11990912119905minus12)10038161003816100381610038161003816+ 119862119905minus12 ⟨11990912119905minus16⟩minus34 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2

le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)

Thuswe need to estimate the norm 120597120585119908L2 = J119910L2 For thedifference 119910 we get from (1) L120597minus1119909 P119910 = 120582(P + 2)(|1199061|21199061 minus|1199062|21199062)Hence by the energy method

119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909

1003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817119906111990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)

Note that P1199062 = 120597119909119909119905minus13119891(119909119905minus13) + 3119905120597119905119905minus13119891(119909119905minus13) = 0for the case of self-similar solution1199062 = 119905minus13119891(119909119905minus13) Hence

119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)

By (78) we have ⟨11990912119905minus16⟩12|119906(119905 119909)| le 119862120576119905minus13 To estimate|1199061|21199061 minus |1199062|21199062L2 we use the above estimates to get

100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin

100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩minus1 119910100381710038171003817100381710038171003817L2le 1198621205762119905minus23 100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩minus1 119910100381710038171003817100381710038171003817L2

(83)

In view of Lemma 4100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩minus1 119910100381710038171003817100381710038171003817L2le 119862119905minus13 1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩minus32 119908(119905 11990912119905minus12)1003817100381710038171003817100381710038171003817L2119909+ 119862119905minus12 1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩minus741003817100381710038171003817100381710038171003817L2 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2

(84)

Since 119908(0) = 0 we get by the Hardy inequality⟨|119909|12119905minus16⟩minus32119908(119905 11990912119905minus12)L2119909 le 119862120597120585119908L2 and by adirect calculation ⟨|119909|12119905minus16⟩minus74L2 le 11986211990516 Hence


⟨|119909|12119905minus16⟩minus1119910L2 le 119862119905minus13120597120585119908L2 Therefore by (83)|1199061|21199061 minus |1199062|21199062L2 le 1198621205762119905minus1120597120585119908L2 le 1198621205763119905minus1+120574 Thus weobtain from (82) (119889119889119905)120597minus1119909 P119910L2 le 1198621205763119905minus1+120574 which implies120597minus1119909 P119910L2 le 1198621205763119905120574 Therefore J119910L2 le 120597minus1119909 P119910L2 +119862119905|1199061|21199061 minus |1199062|21199062L2 le 1198621205763119905120574 Lemma 14 is proved

Now we turn to the proof of asymptotic formula (7)for the solutions 119906 of Cauchy problem (1) Let V119898(119905 119909) bethe self-similar solution with the total mass condition 119898 =(1radic2120587) intR 1199060(119909)119889119909 = (1radic2120587) intR V119898(119905 119909)119889119909 = 0 Note thatV119898Xinfin le 119862120576 by Theorem 2 and 119906Xinfin le 119862120576 by Theorem 1Also 119898 = 120593(119905 0) = V119898(119905 0) for 119905 ge 1 Then by Lemma 14we find 119906(119905 119909) = V119898(119905 119909) + 119874(120576119905minus12+120574)Thus asymptotics (7)follows Theorem 3 is proved

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

The work is partially supported by CONACYT 252053-F andPAPIIT Project IN100817

References

[1] F J Diaz-Otero and P Chamorro-Posada ldquoInterchannel solitoncollisions in periodic dispersion maps in the presence of thirdorder dispersionrdquo Journal of Nonlinear Mathematical Physicsvol 15 pp 137ndash143 2008

[2] J R Taylor Generation and compression of femtosecond solitonsin optical fibers in Optical Solitons-Theory and Experiment J RTaylor Ed Cambridge Studies in Modern Optics CambridgeUniversity Press Cambridge UK 1992

[3] S Cui and S Tao ldquoStrichartz estimates for dispersive equationsand solvability of the Kawahara equationrdquo Journal ofMathemat-ical Analysis and Applications vol 304 no 2 pp 683ndash702 2005

[4] S B Cui D G Deng and S P Tao ldquoGlobal existence ofsolutions for the Cauchy problem of the Kawahara equationwith L2 initial datardquo Acta Mathematica Sinica vol 22 no 5 pp1457ndash1466 2006

[5] Y Kodama and A Hasegawa ldquoNonlinear pulse propagationin a monomode dielectric guiderdquo IEEE Journal of QuantumElectronics vol 23 no 5 pp 510ndash524 1987

[6] A T Ilrsquoichev and A Y Semenov ldquoStability of solitary waves indispersive media described by a fifth-order evolution equationrdquoTheoretical and Computational Fluid Dynamics vol 3 no 6 pp307ndash326 1992

[7] T Kawahara ldquoOscillatory solitary waves in dispersive mediardquoJournal of the Physical Society of Japan vol 33 no 1 pp 260ndash264 1972

[8] Y Kodama ldquoOptical solitons in a monomode fiberrdquo Journal ofStatistical Physics vol 39 no 5-6 pp 597ndash614 1985

[9] A Mussot A Kudlinski E Louvergneaux M Kolobov and MTaki ldquoImpact of the third-order dispersion on the modulationinstability gain of pulsed signalsrdquo Optics Expresss vol 35 no 8pp 1194ndash1196 2010

[10] I Naumkin and R Weder ldquoHigh-energy and smoothnessasymptotic expansion of the scattering amplitude for the Diracequation and applicationrdquoMathematical Methods in the AppliedSciences vol 38 no 12 pp 2427ndash2465 2015

[11] M Taki AMussot A Kudlinski E LouvergneauxMKolobovand M Douay ldquoThird-order dispersion for generating opticalrogue solitonsrdquo Physics Letters A vol 374 no 4 pp 691ndash6952010

[12] X Carvajal ldquoLocal well-posedness for a higher order nonlinearSchrodinger equation in Sobolev spaces of negative indicesrdquoElectronic Journal of Differential Equations vol 13 pp 1ndash132004

[13] X Carvajal and F Linares ldquoA higher order nonlinearSchrodinger equation with variable coefficientsrdquo DifferentialIntegral Equations vol 16 no 9 pp 1111ndash1130 2003

[14] C Laurey ldquoThe Cauchy problem for a third order nonlinearSchrodinger equationrdquo Nonlinear Analysis Theory Methods ampApplications vol 29 no 2 pp 121ndash158 1997

[15] J L Bona G Ponce J-C Saut and C Sparber ldquoDispersiveblow-up for nonlinear Schrodinger equations revisitedrdquo Journalde Mathematiques Pures et Appliquees vol 102 no 4 pp 782ndash811 2014

[16] J Ginibre Y Tsutsumi and G Velo ldquoExistence and uniquenessof solutions for the generalized Korteweg de Vries equationrdquoMathematische Zeitschrift vol 203 no 1 pp 9ndash36 1990

[17] N Hayashi ldquoAnalyticity of solutions of the Korteweg-de Vriesequationrdquo SIAM Journal on Mathematical Analysis vol 22 no6 pp 1738ndash1743 1991

[18] T Kato ldquoOn the Cauchy problem for the (generalized)Korteweg-deVries equationrdquo in Studies in AppliedMathematicsvol 8 of Adv Math Suppl Stud pp 93ndash128 Academic PressNew York 1983

[19] C E Kenig G Ponce and L Vega ldquoOn the (generalized)Korteweg-de Vries equationrdquo Duke Mathematical Journal vol59 no 3 pp 585ndash610 1989

[20] C E Kenig G Ponce and L Vega ldquoWell-posedness of theinitial value problem for the Korteweg-de Vries equationrdquoJournal of the American Mathematical Society vol 4 no 2 pp323ndash347 1991

[21] C E Kenig G Ponce and L Vega ldquoWell-posedness and scat-tering results for the generalized Korteweg-de Vries equationvia the contraction principlerdquo Communications on Pure andApplied Mathematics vol 46 no 4 pp 527ndash620 1993

[22] S N Kruzhkov A V Faminskiı and J R Schulenberger ldquoGen-eralized Solutions of the Cauchy Problem for the Korteweg-deVries EquationrdquoMathematics of the USSR - Sbornik vol 48 no2 pp 391ndash421 1984

[23] J-C Saut ldquoSur quelques generalizations de lrsquoequation deKorteweg- de Vriesrdquo Journal de Mathematiques Pures etAppliquees vol 58 no 1 pp 21ndash61 1979

[24] G Staffilani ldquoOn the generalized Korteweg-de Vries-type equa-tionsrdquoDifferential Integral Equations vol 10 no 4 pp 777ndash7961997

[25] M Tsutsumi ldquoOn global solutions of the generalized Kortweg-de Vries equationrdquo Publications of the Research Institute forMathematical Sciences vol 7 pp 329ndash344 197172


[26] A De Bouard N Hayashi and K Kato ldquoGevrey regularizingeffect for the (generalized) Korteweg - de Vries equation andnonlinear Schrodinger equationsrdquo Annales de lrsquoInstitut HenriPoincare C Analyse Non Lineaire vol 12 no 6 pp 673ndash7251995

[27] T Cazenave and I Naumkin ldquoModified scattering for thecritical nonlinear Schrodinger equationrdquo Journal of FunctionalAnalysis vol 274 no 2 pp 402ndash432 2018

[28] P Constantin and J-C Saut ldquoLocal smoothing properties ofdispersive equationsrdquo Journal of the American MathematicalSociety vol 1 no 2 pp 413ndash439 1988

[29] W Craig K Kappeler andW A Strauss ldquoGain of regularity forsolutions of KdV typerdquo Annales de lrsquoInstitut Henri Poincare CAnalyse Non Lineaire vol 9 no 2 pp 147ndash186 1992

[30] S Klainerman ldquoLong-time behavior of solutions to nonlinearevolution equationsrdquo Archive for Rational Mechanics and Anal-ysis vol 78 no 1 pp 73ndash98 1982

[31] S Klainerman andG Ponce ldquoGlobal small amplitude solutionsto nonlinear evolution equationsrdquoCommunications on Pure andApplied Mathematics vol 36 no 1 pp 133ndash141 1983

[32] J L Bona and J-C Saut ldquoDispersive blowup of solutions ofgeneralized Korteweg-de Vries equationsrdquo Journal of Differen-tial Equations vol 103 no 1 pp 3ndash57 1993

[33] N Hayashi and E I Kaikina ldquoAsymptotics for the third-order nonlinear Schrodinger equation in the critical caserdquoMathematical Methods in the Applied Sciences vol 40 no 5 pp1573ndash1597 2017

[34] N Hayashi J A Mendez-Navarro and P Naumkin ldquoAsymp-totics for the fourth-order nonlinear Schrodinger equation inthe critical caserdquo Journal of Differential Equations vol 261 no9 pp 5144ndash5179 2016

[35] N Hayashi and P I Naumkin ldquoThe initial value problemfor the cubic nonlinear Klein-Gordon equationrdquo Zeitschrift furAngewandte Mathematik und Physik vol 59 no 6 pp 1002ndash1028 2008

[36] N Hayashi and T Ozawa ldquoScattering theory in the weightedL2(Rn) spaces for some Schr odinger equationsrdquo Annales delrsquoInstitut Henri Poincare Section A Physique Theorique vol 48no 1 pp 17ndash37 1988

[37] I P Naumkin ldquoSharp asymptotic behavior of solutions for cubicnonlinear Schrodinger equations with a potentialrdquo Journal ofMathematical Physics vol 57 no 5 Article ID 051501 05150131 pages 2016

[38] I P Naumkin ldquoInitial-boundary value problem for the onedimensional Thirring modelrdquo Journal of Differential Equationsvol 261 no 8 pp 4486ndash4523 2016

[39] M V Fedoryuk ldquoAsymptotics integrals and seriesrdquo in Math-ematical Reference Library Nauka vol 13 p 544 Springer-Verlag Moscow 1987

[40] E M Stein and R Shakarchi ldquoFunctional analysisrdquo in Introduc-tion to Further Topics in Analysis vol 4 of Princeton Lectures inAnalysis Princeton University Press Princeton NJ USA 2011

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function 120601 then the inverse Fourier transformation is givenbyFminus1120601 = (1radic2120587) int

R119890119894119909120585120601(120585)119889120585

We are now in a position to state our result

Theorem 1 Assume that the initial data 1199060 isin H1 capH01 havea sufficiently small norm 1199060H1capH01 le 120576 Then there exists aunique global solution F119890minus(1198941199053)|120597119909|3119906 isin C([0infin) Linfin cap H01)of Cauchy problem (1) Furthermore the estimate

sup119905gt0

(100381710038171003817100381710038171003817F119890minus(1198941199053)|120597119909|3119906 (119905)100381710038171003817100381710038171003817Linfin+ ⟨119905⟩minus16 100381710038171003817100381710038171003817119909119890minus(1198941199053)|120597119909|3119906 (119905)100381710038171003817100381710038171003817L2+ ⟨119905⟩(13)(1minus1119901) 119906 (119905)L119901) le 119862120576

(2)

is true where 119901 gt 4Next we prove the existence of the self-similar solutions

V119898(119905 119909) = 119905minus13119891119898(119909119905minus13)Theorem 2 There exists a unique solution of Cauchy problem(1) in the self-similar form V119898(119905 119909) = 119905minus13119891119898(119909119905minus13) such that

119898 = 1radic2120587 intR V119898 (119905 119909) 119889119909 = 1radic2120587 intR 119891119898 (119909) 119889119909 = 0 (3)

where119898 is sufficiently small number and

F119890minus(1198941199053)|120597119909|3V119898 isin C ([1infin) Linfin) 119909119890minus(1198941199053)|120597119909|3V119898 isin C ([1infin) L2) (4)

Furthermore the estimate

sup119905gt1

(100381710038171003817100381710038171003817F119890minus(1198941199053)|120597119909|3V119898 (119905)100381710038171003817100381710038171003817Linfin+ 119905minus16 100381710038171003817100381710038171003817119909119890minus(1198941199053)|120597119909|3V119898 (119905)100381710038171003817100381710038171003817L2+ 119905(13)(1minus1119901) 1003817100381710038171003817V119898 (119905)1003817100381710038171003817L119901) le 119862 |119898|

(5)

is true where 119901 gt 4Nowwe state the stability of solutions to Cauchy problem

(1) in the neighborhood of the self-similar solution V119898(119905 119909)Theorem 3 Suppose that

1radic2120587 intR 119891119898 (119909) 119889119909 = 1radic2120587 intR 1199060 (119909) 119889119909 = 119898 = 0 (6)

Let 119906(119905 119909) and V119898(119905 119909) be the solutions constructed in Theo-rems 1 and 2 respectively Then there exists small 120574 gt 0 suchthat the asymptotics

1003817100381710038171003817119906 (119905) minus V119898 (119905)1003817100381710038171003817Linfin le 119862119905minus12+120574 (7)

are true for 119905 ge 1

Our approach is based on the factorization techniquesDefine the free evolution group U(119905) = Fminus1119890minus(1198941199053)|120585|3F andwrite

U (119905)Fminus1120601 = D119905radic |119905|2120587 intR 119890119894119905(119909120585minus(13)|120585|3)120601 (120585) 119889120585 (8)

where D119905120601 = |119905|minus12120601(119909119905) is the dilation operator Thereis a unique stationary point 120585 = 120583(119909) equiv (119909|119909|)radic|119909| inthe integral int

R119890119894119905(119909120585minus(13)|120585|3)120601(120585)119889120585 which is defined as the

root of the equation 120585|120585| = 119909 for all 119909 isin R Define thescaling operator (B120601)(119909) = 120601(120583(119909)) Hence we find thefollowing decomposition U(119905)Fminus1120601 = D119905B119872V120601 wherethe multiplication factor119872 = 119890(21198941199053)|120578|3 and the deformationoperator

V (119905) 120601 = radic |119905|2120587 intR 119890minus119894119905119878(120585120578)120601 (120585) 119889120585 (9)

where the phase function 119878(120585 120578) = (13)|120585|3 minus (13)|120578|3 minus120578|120578|(120585 minus 120578) Denote A119896 = 119872119896(12119905|120578|)120597120578119872119896 119896 = 0 1 WehaveA1 = A0 + 119894120578 and alsoA1V = V119894120585 [119894120578V] = minusA0Vtherefore we obtain the commutator 120597120578V = minus2119905|120578|[119894120578V]Since 120597120585119878(120585 120578) = 120585|120585| minus 120578|120578| then we get 119894119905[120578|120578|V]120601 =minusV120597120585120601 Also we need the representation for the inverseevolution group FU(minus119905)120601 = Vlowast119872Bminus1Dminus1

119905 where theinverse dilation operator Dminus1

119905 120601 = |119905|12120601(119909119905) the inversescaling operator (Bminus1120601)(120578) = 120601(120578|120578|) and the inversedeformation operator

Vlowast (119905) 120601 = radic2 |119905|120587 int

R

119890119894119905119878(120585120578)120601 (120578) 10038161003816100381610038161205781003816100381610038161003816 119889120578 (10)

We have 119894120585Vlowast120601 = VlowastA1120601 Hence the commutator[119894120585Vlowast] = VlowastA0 Define the new dependent variable 120593 =FU(minus119905)119906(119905) Since FU(minus119905)L = 120597119905FU(minus119905) with L = 120597119905 +(1198943)|120597119909|3 applying the operator FU(minus119905) to (1) substituting119906(119905) = U(119905)Fminus1120593 = D119905B119872V120593 and using the factorizationtechniques we get

120597119905120593 = FU (minus119905)L119906 = 119894120582120585FU (minus119905) (|119906|2 119906)= 119894120582120585Vlowast119872B

minus1D

minus1119905 (1003816100381610038161003816D119905B119872V12059310038161003816100381610038162 D119905B119872V120593)

= 119894120582120585119905minus1Vlowast119872Bminus1 (1003816100381610038161003816B119872V12059310038161003816100381610038162 B119872V120593)

= 119894120582120585119905minus1Vlowast119872(1003816100381610038161003816119872V12059310038161003816100381610038162119872V120593)= 119894120582120585119905minus1Vlowast (1003816100381610038161003816V12060110038161003816100381610038162 V120601)

(11)

since the nonlinearity is gauge invariant Finally we mentionsome important identities The operator J = U(119905)119909U(minus119905) =119909 + 119894119905120597119909|120597119909| plays a crucial role in the large time asymptoticestimates Note thatJ commutes withL that is [JL] = 0To avoid the derivative loss we also use the operator P =3119905120597119905 + 120597119909119909 Note the commutator relation [P 119890minus(1198941199053)|120585|3] = 0









intR

119890119894119911119892(119910)119891 (119910) 119889119910= 119890119894119911119892(1199100)119891 (1199100)radic 2120587119911 100381610038161003816100381611989210158401015840 (1199100)1003816100381610038161003816 119890119894(1205874)sgn11989210158401015840(1199100)

+ 119874 (119911minus32)(13)


119860119895 (119905 120578) = 11990512 10038161003816100381610038161205781003816100381610038161003816 120578119895radic2119894 ⟨1199051205783⟩ + 119874(119905121205781+119895 ⟨1199051205783⟩minus1) (14)



119890119894119905119878(120585120578)Θ(120578120585minus1) 10038161003816100381610038161205781003816100381610038161003816 119889120578= 1205852radic2 |119905|120587 int3

13119890119894119905|120585|3119866(119910)Θ(119910) 10038161003816100381610038161199101003816100381610038161003816 119889119910

(15)







(18)




R


R


(20)


(21)


13lt120585120578lt3


+ 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 int13lt120585120578lt3

1003816100381610038161003816120585 minus 1205781003816100381610038161003816 10038161003816100381610038161003816120597120585120601 (120585)10038161003816100381610038161003816 1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2 (22)




100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int

13lt120585120578lt3

(120585 minus 120578)2 119889120585(1 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2)2)

12

le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895+32 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int3

13

(119910 minus 1)2 119889119910(1 + |119905| 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2)2)

12

le 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 10038161003816100381610038161205781003816100381610038161003816119895+32 ⟨1199051205783⟩minus34

(23)


119890minus119894119905119878(120585120578) = 1198672120597120585 (120585119890minus119894119905119878(120585120578)) (24)


sdot intR


R


R


(25)


le 119862 100381610038161003816100381612058510038161003816100381610038161+1198951 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782 (26)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int|120585|le3|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|le3|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 1199051205851205782)2)12

(27)

We have int|120585|le3|120578|

(|120585|119895119889120585(1 + 119905|120585|1205782)) le 119862|120578|119895+1 int3


|120585|le3|120578|(|120585|2+2119895119889120585

(1 + 119905|120585|1205782)2) le 119862|120578|2119895+3 int3



V2120585119895120601 = 11986211990512120601 (0)sdot int

R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R


R

119890minus119894119905119878(120585120578)1205851+1198951205942 (120585120578minus1)1198672120601120585 (120585) 119889120585

(28)


10038161003816100381610038161003816V212058511989512060110038161003816100381610038161003816le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int

|120585|ge(32)|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 100381610038161003816100381612058510038161003816100381610038163+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|ge(32)|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 119905 100381610038161003816100381612058510038161003816100381610038163)2)12

(29)

We have int|120585|ge(32)|120578|

(|120585|119895119889120585(1+119905|120585|3)) le 119862|120578|1+119895 intinfin


|120585|ge(32)|120578|(|120585|2+2119895119889120585(1 +119905|120585|3)2) le 119862|120578|3+2119895 intinfin











= 1198681 + 1198682

(31)



1198681 = 11986211990512 intinfin



+ 11986211990512 intinfin



(33)


10038171003817100381710038171003817100381710038171003817120581 (119909)119909 (120601 (119909) minus 120601 (0))10038171003817100381710038171003817100381710038171003817L2 =10038171003817100381710038171003817100381710038171003817120581 (119909)119909 int119909

01206011015840 (119910) 11988911990511991010038171003817100381710038171003817100381710038171003817L2

= 100381710038171003817100381710038171003817100381710038171003817120581 (119909) int1

01206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le 119862 100381710038171003817100381710038171003817100381710038171003817int1

0120581 (119909119910) 1206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le int1

0

10038171003817100381710038171003817120581 (119909119910) 1206011015840 (119909119910)10038171003817100381710038171003817L2 119889119905 le int1

0119910minus12119889119910 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

le 2 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

(34)



13le120578120585le3

(120578 minus 120585)2 119889120578(1 + 119905 10038161003816100381610038161205851003816100381610038161003816 (120578 minus 120585)2)2)

12


13

100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2 119889119910(1 + 119905 100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2)2)12

le 11986211990512 1003816100381610038161003816120585100381610038161003816100381652minus120572


(35)



1198682 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)


minusinfin119890119894119905119878(120585120578)1205781198674 (1


(37)



(38)



minusinfin

⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2)12 le 11986211990512 10038171003817100381710038171206011003817100381710038171003817I120572120573

sdot (intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2 )12

(39)


We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1


+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513


le 119862119905(2120572minus5)3

(40)


3 Commutators withV


W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)


R


R


(42)




R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)



R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)


0


1


0


1






(45)





(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018
















intR

119890119894119911119892(119910)119891 (119910) 119889119910= 119890119894119911119892(1199100)119891 (1199100)radic 2120587119911 100381610038161003816100381611989210158401015840 (1199100)1003816100381610038161003816 119890119894(1205874)sgn11989210158401015840(1199100)

+ 119874 (119911minus32)(13)


119860119895 (119905 120578) = 11990512 10038161003816100381610038161205781003816100381610038161003816 120578119895radic2119894 ⟨1199051205783⟩ + 119874(119905121205781+119895 ⟨1199051205783⟩minus1) (14)



119890119894119905119878(120585120578)Θ(120578120585minus1) 10038161003816100381610038161205781003816100381610038161003816 119889120578= 1205852radic2 |119905|120587 int3

13119890119894119905|120585|3119866(119910)Θ(119910) 10038161003816100381610038161199101003816100381610038161003816 119889119910

(15)







(18)




R


R


(20)


(21)


13lt120585120578lt3


+ 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 int13lt120585120578lt3

1003816100381610038161003816120585 minus 1205781003816100381610038161003816 10038161003816100381610038161003816120597120585120601 (120585)10038161003816100381610038161003816 1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2 (22)




100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int

13lt120585120578lt3

(120585 minus 120578)2 119889120585(1 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2)2)

12

le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895+32 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int3

13

(119910 minus 1)2 119889119910(1 + |119905| 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2)2)

12

le 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 10038161003816100381610038161205781003816100381610038161003816119895+32 ⟨1199051205783⟩minus34

(23)


119890minus119894119905119878(120585120578) = 1198672120597120585 (120585119890minus119894119905119878(120585120578)) (24)


sdot intR


R


R


(25)


le 119862 100381610038161003816100381612058510038161003816100381610038161+1198951 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782 (26)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int|120585|le3|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|le3|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 1199051205851205782)2)12

(27)

We have int|120585|le3|120578|

(|120585|119895119889120585(1 + 119905|120585|1205782)) le 119862|120578|119895+1 int3


|120585|le3|120578|(|120585|2+2119895119889120585

(1 + 119905|120585|1205782)2) le 119862|120578|2119895+3 int3



V2120585119895120601 = 11986211990512120601 (0)sdot int

R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R


R

119890minus119894119905119878(120585120578)1205851+1198951205942 (120585120578minus1)1198672120601120585 (120585) 119889120585

(28)


10038161003816100381610038161003816V212058511989512060110038161003816100381610038161003816le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int

|120585|ge(32)|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 100381610038161003816100381612058510038161003816100381610038163+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|ge(32)|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 119905 100381610038161003816100381612058510038161003816100381610038163)2)12

(29)

We have int|120585|ge(32)|120578|

(|120585|119895119889120585(1+119905|120585|3)) le 119862|120578|1+119895 intinfin


|120585|ge(32)|120578|(|120585|2+2119895119889120585(1 +119905|120585|3)2) le 119862|120578|3+2119895 intinfin











= 1198681 + 1198682

(31)



1198681 = 11986211990512 intinfin



+ 11986211990512 intinfin



(33)


10038171003817100381710038171003817100381710038171003817120581 (119909)119909 (120601 (119909) minus 120601 (0))10038171003817100381710038171003817100381710038171003817L2 =10038171003817100381710038171003817100381710038171003817120581 (119909)119909 int119909

01206011015840 (119910) 11988911990511991010038171003817100381710038171003817100381710038171003817L2

= 100381710038171003817100381710038171003817100381710038171003817120581 (119909) int1

01206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le 119862 100381710038171003817100381710038171003817100381710038171003817int1

0120581 (119909119910) 1206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le int1

0

10038171003817100381710038171003817120581 (119909119910) 1206011015840 (119909119910)10038171003817100381710038171003817L2 119889119905 le int1

0119910minus12119889119910 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

le 2 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

(34)



13le120578120585le3

(120578 minus 120585)2 119889120578(1 + 119905 10038161003816100381610038161205851003816100381610038161003816 (120578 minus 120585)2)2)

12


13

100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2 119889119910(1 + 119905 100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2)2)12

le 11986211990512 1003816100381610038161003816120585100381610038161003816100381652minus120572


(35)



1198682 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)


minusinfin119890119894119905119878(120585120578)1205781198674 (1


(37)



(38)



minusinfin

⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2)12 le 11986211990512 10038171003817100381710038171206011003817100381710038171003817I120572120573

sdot (intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2 )12

(39)


We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1


+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513


le 119862119905(2120572minus5)3

(40)


3 Commutators withV


W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)


R


R


(42)




R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)



R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)


0


1


0


1






(45)





(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018











100381610038161003816100381611986811003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int

13lt120585120578lt3

(120585 minus 120578)2 119889120585(1 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2)2)

12

le 11986211990512 10038161003816100381610038161205781003816100381610038161003816119895+32 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2sdot (int3

13

(119910 minus 1)2 119889119910(1 + |119905| 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2)2)

12

le 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 10038161003816100381610038161205781003816100381610038161003816119895+32 ⟨1199051205783⟩minus34

(23)


119890minus119894119905119878(120585120578) = 1198672120597120585 (120585119890minus119894119905119878(120585120578)) (24)


sdot intR


R


R


(25)


le 119862 100381610038161003816100381612058510038161003816100381610038161+1198951 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782 (26)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int|120585|le3|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|le3|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 1199051205851205782)2)12

(27)

We have int|120585|le3|120578|

(|120585|119895119889120585(1 + 119905|120585|1205782)) le 119862|120578|119895+1 int3


|120585|le3|120578|(|120585|2+2119895119889120585

(1 + 119905|120585|1205782)2) le 119862|120578|2119895+3 int3



V2120585119895120601 = 11986211990512120601 (0)sdot int

R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 120585119895) 119889120585+ 11986211990512 int

R


R

119890minus119894119905119878(120585120578)1205851+1198951205942 (120585120578minus1)1198672120601120585 (120585) 119889120585

(28)


10038161003816100381610038161003816V212058511989512060110038161003816100381610038161003816le 11986211990512 1003816100381610038161003816120601 (0)1003816100381610038161003816 int

|120585|ge(32)|120578|

10038161003816100381610038161205851003816100381610038161003816119895 1198891205851 + 119905 100381610038161003816100381612058510038161003816100381610038163+ 11986211990512 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (int|120585|ge(32)|120578|

100381610038161003816100381612058510038161003816100381610038162+2119895 119889120585(1 + 119905 100381610038161003816100381612058510038161003816100381610038163)2)12

(29)

We have int|120585|ge(32)|120578|

(|120585|119895119889120585(1+119905|120585|3)) le 119862|120578|1+119895 intinfin


|120585|ge(32)|120578|(|120585|2+2119895119889120585(1 +119905|120585|3)2) le 119862|120578|3+2119895 intinfin











= 1198681 + 1198682

(31)



1198681 = 11986211990512 intinfin



+ 11986211990512 intinfin



(33)


10038171003817100381710038171003817100381710038171003817120581 (119909)119909 (120601 (119909) minus 120601 (0))10038171003817100381710038171003817100381710038171003817L2 =10038171003817100381710038171003817100381710038171003817120581 (119909)119909 int119909

01206011015840 (119910) 11988911990511991010038171003817100381710038171003817100381710038171003817L2

= 100381710038171003817100381710038171003817100381710038171003817120581 (119909) int1

01206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le 119862 100381710038171003817100381710038171003817100381710038171003817int1

0120581 (119909119910) 1206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le int1

0

10038171003817100381710038171003817120581 (119909119910) 1206011015840 (119909119910)10038171003817100381710038171003817L2 119889119905 le int1

0119910minus12119889119910 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

le 2 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

(34)



13le120578120585le3

(120578 minus 120585)2 119889120578(1 + 119905 10038161003816100381610038161205851003816100381610038161003816 (120578 minus 120585)2)2)

12


13

100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2 119889119910(1 + 119905 100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2)2)12

le 11986211990512 1003816100381610038161003816120585100381610038161003816100381652minus120572


(35)



1198682 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)


minusinfin119890119894119905119878(120585120578)1205781198674 (1


(37)



(38)



minusinfin

⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2)12 le 11986211990512 10038171003817100381710038171206011003817100381710038171003817I120572120573

sdot (intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2 )12

(39)


We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1


+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513


le 119862119905(2120572minus5)3

(40)


3 Commutators withV


W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)


R


R


(42)




R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)



R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)


0


1


0


1






(45)





(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018














= 1198681 + 1198682

(31)



1198681 = 11986211990512 intinfin



+ 11986211990512 intinfin



(33)


10038171003817100381710038171003817100381710038171003817120581 (119909)119909 (120601 (119909) minus 120601 (0))10038171003817100381710038171003817100381710038171003817L2 =10038171003817100381710038171003817100381710038171003817120581 (119909)119909 int119909

01206011015840 (119910) 11988911990511991010038171003817100381710038171003817100381710038171003817L2

= 100381710038171003817100381710038171003817100381710038171003817120581 (119909) int1

01206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le 119862 100381710038171003817100381710038171003817100381710038171003817int1

0120581 (119909119910) 1206011015840 (119909119910) 119889119910100381710038171003817100381710038171003817100381710038171003817L2

le int1

0

10038171003817100381710038171003817120581 (119909119910) 1206011015840 (119909119910)10038171003817100381710038171003817L2 119889119905 le int1

0119910minus12119889119910 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

le 2 10038171003817100381710038171003817120581120601101584010038171003817100381710038171003817L2

(34)



13le120578120585le3

(120578 minus 120585)2 119889120578(1 + 119905 10038161003816100381610038161205851003816100381610038161003816 (120578 minus 120585)2)2)

12


13

100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2 119889119910(1 + 119905 100381610038161003816100381612058510038161003816100381610038163 (1 minus 119910)2)2)12

le 11986211990512 1003816100381610038161003816120585100381610038161003816100381652minus120572


(35)



1198682 = 11986211990512 intinfin

minusinfin119890119894119905119878(120585120578)


minusinfin119890119894119905119878(120585120578)1205781198674 (1


(37)



(38)



minusinfin

⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2)12 le 11986211990512 10038171003817100381710038171206011003817100381710038171003817I120572120573

sdot (intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2 )12

(39)


We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1


+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513


le 119862119905(2120572minus5)3

(40)


3 Commutators withV


W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)


R


R


(42)




R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)



R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)


0


1


0


1






(45)





(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018









We have

intinfin

minusinfin

⟨120585⟩2120573 ⟨120578⟩2120573 100381610038161003816100381612057810038161003816100381610038164minus2120572 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2le 119862int1

0

⟨120585⟩2120573 1205784minus2120572 ⟨120578⟩2120573 119889120578(1 + 1199051205782 (10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816))2

+ 11986211990521205733minus2 intinfin

1⟨120585⟩2120573 1205782120573minus2120572119889120578(10038161003816100381610038161205851003816100381610038161003816 + 10038161003816100381610038161205781003816100381610038161003816)2

le 119862int1

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905minus41205733 int1


+ 11986211990541205733minus2 intinfin

11205784120573minus2120572minus2119889120578

le 119862119905(2120572minus5)3 int11990513

01205784minus2120572 ⟨120578⟩2120573minus6 119889120578

+ 119862119905(2120572minus5)3 int11990513


le 119862119905(2120572minus5)3

(40)


3 Commutators withV


W120601 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 120601 (120585) 119889120585 (41)


R


R


(42)




R

119889120577119890119894119905(119878(120577120578)minus119878(120585120578))119902 (119905 120585 120578) 120601 (120585) 119902 (119905 120577 120578)120601 (120577)= 119862119905int

R

119889120585119890minus(1198941199053)|120585|3120601 (120585)sdot int

R

119889120577119890(1198941199053)|120577|3120601 (120577)119870 (119905 120585 120577)

(43)



R

1198901198941199051205782(120585minus120577)120578120597120578 (119867120578120597120578 (119867 10038161003816100381610038161205781003816100381610038161003816 119902 (119905 120585 120578)sdot 119902 (119905 120577 120578))) 119889120578 (44)


0


1


0


1






(45)





(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018










(46)




W1 = 11990512 intR

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 119889120585= 11990512 int

R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578)Θ (120585120578minus1) 119889120585+ 11990512 int

R


R

119890minus119894119905119878(120585120578)119902 (119905 120585 120578) 1205942 (120585120578minus1) 119889120585 = 1198681+ 1198682 + 1198683

(47)


1198681 = 11986211990512 intR



13lt120585120578lt3

1198891205851 + 119905 10038161003816100381610038161205781003816100381610038161003816 (120585 minus 120578)2le 11986211990512 int3

13

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 (119910 minus 1)2 le 11986211990512 ⟨120578⟩minus32 (49)


1198682 = 11986211990512 intR


(50)


100381610038161003816100381611986821003816100381610038161003816 le 11986211990512 10038161003816100381610038161205781003816100381610038161003816minus1 int|120585|le3|120578|

1198891205851 + 119905 10038161003816100381610038161205851003816100381610038161003816 1205782le 11986211990512 int3

0

1198891199101 + 119905 100381610038161003816100381612057810038161003816100381610038163 119910 le 11986211990512 ⟨120578⟩minus32 (51)


R

119890minus119894119905119878(120585120578)120585120597120585 (11986721205942 (120585120578minus1) 119902 (119905 120585 120578)) 119889120585 (52)


(32)|120578|

119889120585(10038161003816100381610038161205781003816100381610038161003816 + 10038161003816100381610038161205851003816100381610038161003816) (1 + 119905 100381610038161003816100381612058510038161003816100381610038163)le 11986211990512 int(32)⟨120578⟩|120578|minus1

32


(32)⟨120578⟩120585minus3119889120585


(53)



(54)






(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018










(55)


= 119874 (120578] ⟨120578⟩minus]) (56)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199021 (119905 120585 120578) 120597120585120601 (120585) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2

(57)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (58)

Also we have


(59)



proved





(60)





le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (62)



10038171003817100381710038171003817100381710038171003817L2 le 119862 1003817100381710038171003817100381712059712058512060110038171003817100381710038171003817L2 (63)


100381710038171003817100381710038171003817100381710038171003817100381710038171003816100381610038161003816120578100381610038161003816100381612radic |119905|2120587 intR 119890minus119894119905119878(120585120578)1199023 (119905 120585 120578) 119889120585

10038171003817100381710038171003817100381710038171003817100381710038171003817L2 le 11986211990516 (64)

Lemma 9 is proved





(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018











(65)






119906X119879 = sup119905isin[1119879]


(66)





(67)


lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595 = 119860lowastA1

100381610038161003816100381612059510038161003816100381610038162 120595+ 119874(119905]3minus16 ⟨120585⟩minus2]


(68)



100381710038171003817100381710038171003817L2+ 119862 1003817100381710038171003817100381710038161003816100381610038161205781003816100381610038161003816] 120578minus] ⟨120578⟩minus] 1205951205951


1003817100381710038171003817Linfin)

(69)


100381710038171003817100381710038171003817L2le 11986211990516 10038171003817100381710038171205931003817100381710038171003817Y (70)









1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018















1003816100381610038161003816120593 (119905 120585)10038161003816100381610038162 le 10038161003816100381610038161003816120593 (120585minus3 120585)100381610038161003816100381610038162+ 119862 100381710038171003817100381712059310038171003817100381710038174Y int119905

120585minus312058512059113 ⟨12058512059113⟩minus] 119889120591120591

le 1205762 + 1198621205764 int12058511990513

1⟨119910⟩minus1minus] 119889119910 le 1205762 + 1198621205764

(72)






(74)










120597120578120601119898119877 (120578) = 3119894120582VlowastΘ119877

10038161003816100381610038161003816VΘ119877120601119898119877

100381610038161003816100381610038162 VΘ119877120601119898119877

equiv 119865119877 (120601119898119877 (120578)) (76)


+ 119874 (⟨120578⟩minus1minus] 100381710038171003817100381712060111989811987710038171003817100381710038173Z) (77)


0⟨120578⟩minus1minus1205741206011198981198773Z119889120578)







le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018













le 119862119905minus12 1003817100381710038171003817100381712059712058511990810038171003817100381710038171003817L2 (78)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2= 4Reint

R120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

+ 2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

+ 4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 1199062) 119889119909

(79)

Next we get

4ReintR120597minus1119909 P119910 (10038161003816100381610038161199061

10038161003816100381610038162 P1199061 minus 1003816100381610038161003816119906210038161003816100381610038162 P1199062) 119889119909

= 4ReintR

1003816100381610038161003816119906110038161003816100381610038162 120597minus1119909 P119910P119910119889119909

+ 4ReintR(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

= 4ReintR120597119909 10038161003816100381610038161199061

10038161003816100381610038162 10038161003816100381610038161003816120597minus1119909 P119910100381610038161003816100381610038162 119889119909+ 4Reint

R(10038161003816100381610038161199061

10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062120597minus1119909 P119910119889119909

le 119862 1003817100381710038171003817100381610038161003816100381611990611003816100381610038161003816 1199061119909


10038161003816100381610038162 minus 1003816100381610038161003816119906210038161003816100381610038162)P1199062

10038171003817100381710038171003817L2 (80)

In the same manner

2ReintR120597minus1119909 P119910 (1199062

1P1199061 minus 11990622P1199062) 119889119909

= 2ReintR11990621120597minus1119909 P119910P119910119889119909

+ 2ReintR(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909

= 2ReintR1205971199091199062

1120597minus1119909 P1199102 119889119909+ 2Reint

R(1199062

1 minus 11990622)P1199062120597minus1119909 P119910119889119909


+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 10038171003817100381710038171003817(11990621 minus 1199062

2)P1199062

10038171003817100381710038171003817L2

(81)


119889119889119905 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2le 119862 100381710038171003817100381710038161003816100381610038161199061

1003816100381610038161003816 11990611199091003817100381710038171003817Linfin 10038171003817100381710038171003817120597minus1119909 P119910100381710038171003817100381710038172L2

+ 119862 10038171003817100381710038171003817120597minus1119909 P11991010038171003817100381710038171003817L2 100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2 (82)


100381710038171003817100381710038171003816100381610038161003816119906110038161003816100381610038162 1199061 minus 10038161003816100381610038161199062

10038161003816100381610038162 1199062

10038171003817100381710038171003817L2le 119862 2sum

119895=1

1003817100381710038171003817100381710038171003817⟨|119909|12 119905minus16⟩12 119906119895

10038171003817100381710038171003817100381710038172

Linfin


(83)


(84)





Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018











Data Availability




Acknowledgments


References

















































Journal of












Journal of







Volume 2018






Volume 2018































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








Documents

ResearchArticle High-Speed Transmission in Long …downloads.hindawi.com/journals/ijde/2018/8236942.pdfHigh-Speed Transmission in Long-Haul Electrical Systems BeatrizJuárez-Campos,1