Tarea1versionfinal1

7/25/2019 Tarea1versionfinal1

1/17

D(R) S(R)


2/17

1

4xex

2

cos(ex2

) R

R

1

(1 + |x|2

)

=

x18

sin(ex2

)

18

sin(ex2

)

(Tf) =Tf

18

sin(ex2

)

14

xex2

cos(ex2

)

18

sin(ex2

)

S(R)

| < 18

sin(ex2

), > | 18

R

| sin(ex2)(x)|dx 18

R

|(x)|dx

pN,m (1 + |x|2) 1

8R

1

(1 + |x|2)(1 +

|x|2)

|(x)

|dx

1

8R

1

(1 + |x|2)P0,1() =

1

8P0,1()

R

1

(1 + |x|2) =

8P0,1()

< 1

8sin(ex

2

), > | 8

P0,1()

18

sin(ex2

)

sin(nx)x

D

(R

sin(nx)x

0

R

sin(x)x

=

D(R

< sin(nx)x

, (x)>< 0, (x)>=(0)

| < sin(nx)x

, (x)> (0)| 0

| < sin(nx)x

, (x)> (0)| = |(R

sin(nx)

x ((x))dx) (0)||

|(R

sin(nx)

x ((x))dx) (0)| |

R/[r,r]

sin(nx)

x (x)dx| + |

[r,r]

sin(nx)

x ((x) (0))dx| + |(

[r,r]

sin(nx)

x (0)dx) (0)|

r > 0

> 0

n0 n r

0

0

(x)x

R/[r, r] 1x

|R/[r,r]

sin(nx)(x)

x dx| = |

R

sin(nx)(x)

x 1R/[r,r]dx| 0

n1 n > n1 |

sin(nx)(x)x

| < 3

[r,r]

sin(nx)

x (0)dx= (0)

[r,r]

sin(nx)

xn ndx


3/17

u= nx

(0)

[nr,nr]

sin(u)

u du= (0)

R

sin(u)

u 1[nr,nr]du

sin(u)

u 1[nr,nr]

sin(u)

u

| sin(u)u

1[nr,nr]| |sin(u)

u | L1

(0)

R

sin(u)

u du= (0)

n2 n |[r,r]

sin(nx)x

(0)dx (0)| < 3

| [r,r] sin(nx)x ((x) (0))dx| n r > 0 |(x) (0)| 2n x

[r, r]

n

n max{n1, n2} 3

r < 12

[r,r] |

sin(nx)x

||(x) (0)|dx sin(nx)x

x

0

n

[r, r]

| sin(nx)x

| n

x

sin(nx)

0

|[r,r]

sin(nx)

x ((x) (0))dx|

[r,r]

n

3n

3

| [r,r] sin(nx)x ((x) (0))dx| 3 > 0 n0 n

n1 n2

r

3

n

n max{n1, n2}

>0

n0 n

| < sin(nx)x

, (x)> (0)| 3

+

3 +

3 =

| < sin(nx)x

, (x)> (0)| 0

< sin(nx)

x , (x)>< 0, (x)>=(0)

sin(nx)x

0

D(R).

fn(x) := cos(nx) L2(0, 1) f2n D(0, 1)

< f2n, > < fn, >L2 fn

D

L2

fn L2(0, 1) g L2

| < fn, g > | = | 10

cos(nx)g(x)dx|

u= x

| < fn, g > | = 1

| 0

cos(nu)g(u

)du| < 1

|R

cos(nu)g(u

)1[0,](u)du| 0

g(x)

L2(0, 1)

g( x

)

L2(0, )

g( x

)

L1(0, )

g( x

)1[0,] L1.

fn L2(0, 1) D(R)

< f2n, >=

(cos(nx))2(x)dx=

1

2

10

(x)dx+1

2

10

cos(2nx)(x)dx

u= 2x

< f2n, >= 1

2

10

(x)dx+ 1

4

20

cos(nu)( u

2)du=

1

2

10

(x)dx+ 1

4

R

cos(nu)( u

2)1[0,2](u)du


4/17

< f2n, > 1

2

10

(x)dx

0

f2n 0

12

< f2n, > 1

2

10

(x)dx=

10

1

2(x)dx=

1

x+iy2+z2

xu= u2 D(R3)

Tf D(R3)

f

L1loc(R3)

K

R3

0

K

f

Kf < K

0

K= (, )3 (K/(, )3)

|

1

x+iy2 +z2

dxdydz|

| 1x+iy

2 +z2|dxdydz =

| x i

y2 +z2

x2 +y2 +z2 |dxdydz

x2 +y2 +z2 =r2

20

0

0

| r cos() ir sin()r2

|r2 sin()ddrd = 20

0

0

|r cos() ir sin()| sin()ddrd <

f

L1loc(R3)

Tf

(Tf) =Tf D(R

3)

< x(Tf), >= < Tf, x >= R3

x

x+i

y2 +z2d =

R

R

R

x

x+i

y2 +z2dxdydz

R2

( (x)x+i

y2 +z2

)x=x=dydz+R3

(x)x( 1x+i

y2 +z2

)dxdydz

0

x( 1

x+i

y2 +z2) =x((x+i

y2 +z2)1) = x(x)

(x+i

y2 +z2)2= 1

(x+i

y2 +z2)2

< x(Tf), >= R3

(x) 1

(x+i

y2 +z2)2dxdydz =

x(Tf) = 1

(x+iy2+z2)2

x( 1

(x+iy2 +z2)) =

1

(x+iy2 +z2)2


5/17

D(R) S(R)

D(Rd) C(Rd) f L(Rd)

pN(f) := max||NsupxRd |f(x)|

D(Rd)

(fn) D(Rd) S(Rd) N N, PN(fm fn) 0

NN, >0, n0(); m, n n0(PN(fm fn)

NN, >0, n0(); m, n n0 max||N

supxRd

|fm(x) fn(x)|

N N, > 0, n0(); m, n n0 max||N

||fm(x) fn(x)||

N N, > 0, n0(); m, n n0, || N; ||fm(x) fn(x)||

N

N N, || N, >0, n0(); m, n n0; ||fm(x) fn(x)||

NN, || N; ||fm(x) fn(x)|| 0

fn f L N = 1

L

fm f f C(Rd) S(Rd)

fn C1 fn

C1

fn f fn

f

f

D(Rd)

D(Rd)

D(Rd) D(Rd) D(Rd)

D(Rd)

= C00 C00

0

S(Rd)

11+|x|2

D(Rd) S(Rd) C00 D(Rd)

f C00 fn(x) =f(x)( xn ) D(R)

(x) = 1

|x| < 12

(x) = 0

|x| > 1

|||| = 1 [1, 1]d

fn(x) [n, n]d

f

[n2

, n2

]d

fn D(Rd)

( xn

)

n

f

N

pN(fn f) = max||N

supxRd

|fn(x) f(x)|

|| N ||fn f|| 0

f

C00 0 > 0 f(x)

||x||0

n0 N ||x|| > n0 ||f(x)|| < 2CN2PN,0() || N C

pN,0() = max||N

||||

n 2n0, fn fn f [n0, n0]d

||fn f||= sup||x||n0

|fn(x) f(x)| sup||x||n0

|(f(x)( xn

))| + |f(x)|


6/17

|(f(x)( xn

))| = |

||||

C,f(x)(

x

n)

1

n|

||||

|C,||f(x)||(x

n)| 1

n

C= max||||N

C, N |( xn )| max||N

|||| = pN,0()

|||| |C,||f(x)||(x

n

)

| 1

n N(N 1)C

2

1

npN,0() max

||N

sup||x||n0 |

f(x)

||

sup||x||n0

|(f(x)( xn

))| N(N 1)C2

1

npN,0() max

||Nsup

||x||n0

|f(x)|| N(N 1)C2

pN,0()

n

2CN2pN,0()

2

||fn f|| 2

+

2CN2pN,0()

2+

2 =

>0

n0 n n0 ||fn f||< ||fn f|| n

0

pN(fn f) = max||N

supxRd

|fn(x) f(x)| 0

N

fn f

g

C00 0 0

L

gn D(Rd) gn g

gn n Cn > 0 |x| > C gn(x) = 0 lm|x|

g(x) = L = 0

= L2

n

pN(g gn) supxRd

|gn(x) g(x)| sup|x|>Cn

|gn(x) g(x)| = sup|x|>Cn

|g(x)| L > L2

=

pN(g gn)> n 0 gn g.

h

C00

hn D(Rd) hn h hn n

Cn> 0

|x| > C

hn(x) = 0

N

pN(h hn) supxRd

|hn(x) h(x)| sup|x|>Cn

|hn(x) h(x)| = sup|x|>Cn

|h(x)|

xm Rd |xm| Cn m |xm|

sup|x|>Cn

|h(x)| lm supn

|h(xn)| =M >0

h

0

0 = lm supn

|h(xn)| lm infn

|h(xn)| 0

= mn{M, 1} >0 pN(h hn) > n hn

h

D(Rd) =C00 D(Rd) D(Rd)

D(Rd) S(Rd)

D(Rd)

LP(Rd)

I

I

e.v.t.l.c

I

n D(Rd) n D(Rd) 0 I(n) S(R

d) 0 n D(Rd) nD(Rd) 0

K

n DK n 0


7/17

N, m

pN,m(I(n)) 0 S(Rd)

pN,m(I(n)) = max||N

supxRd

(1 + |x|2)m|n(x)|

K

pN,m(I(n)) = max||N

supxK

(1 + |x|2)m|n(x)| max||N

||n||supxK

(1 + |x|2)m

pN,m(I(n)) = max||N||

n||supxK(1 + |x|2

)m

max||N||

n||Cm

m

0

pN,m(I(n)) max||N

||n||Cm n

0

N

m

I(n) 0 I

D(Rd) S(Rd) D(Rd)

[n, n]

0

D(Rd)

n= 1[0,n](|x|)exp

1| xn |

21

1n

n x n | xn |2 1 0

0

1[0,n](|x|)exp

1| xn |

21

n

n S(Rd) N m pN,m(n) 0

pN,m(n) = max||N

supxRd

(1 + |x|2)m|xn| = max||N

sup|x| n

|x| =n

max||N

sup|x|


8/17

x

n

max||N

sup|x|

lmn

< Tn, >< ex, >

Tn ex


9/17

Tn S(R) Tn f f S(R) S(R) D(R) f D(R) Tn f S(R)

lmn

< Tn, >< f, >

D(R) S(R) D(R)

lmn

< Tn, >< f, >

D(R)

Tn(x) ex

f(x) = ex

ex

S(R)

Tn S(R)

( 1x

) S(R

R

sin(x)

x =

S(R)

< F(vp( 1x

)), >=< v p(1

x), >= lm

0

() ()

d

() =

R

(x)eixdx=

(x)eixdx

< F(vp( 1x

)), >= lm0

(x)e

ixdx (x)eixdx

d

lm0

(x)eix (x)eix

dxd

H(x, ) = (x)eix(x)eix

RR

H(x, )1[,R]()dA

H(x, )1[,R]() (x) |H(x, )1[,R]()|

x

|H(x, )1[,R]()|

= 0

Rx R H(x, )1[,R]()

x

(x)

R

eix eix

ddx

eix = cos(x) i sin(x)

eix = cos(x) +i sin(x)

eix eix = 2i sin(x)

2i(x) R

sin(x)

ddx

T.C.D.

R

2i(x) R

sin(x)

d

R2i(x)

sin(x)

d


10/17

| 2i(x) R

sin(x)

| 2|(x)||

sin(x)

| 2|(x)| L1x

lmR

2i(x) R

sin(x)

ddx=

2i(x)

sin(x)

ddx

lmR

RR

H(x, )1[,R]()dA=

2i(x)

sin(x)

ddx

H

y= x

R

(y

)

eiy eiy

2

dyd=

(y

)

eiy eiy

2

dy

1[,R]()d

(y

)

eiy eiy

2

dy

1[,R]()

(y

)

eiy eiy

2

dy

1[,]()

|

(

y

)

eiy eiy

2

dy

1[,R]()| 2|

(y

)

1

2

dy

1[,]()| 2

1

21[,]()|

(y

)dy

|

12

( y

)dx

lmR

R

(x)

eix eix

dxd= lm

R

R

(

y

) eiy eiy

2

dyd=

(

y

) eiy eiy

2

dyd

(y

)

eiy eiy

2

dyd=

(x)

eix eix

dxd

lmR

RR

H(x, )1[,R]()dA=

(x)

eix eix

dxd

(x)

eix eix

dxd =

2i(x)

sin(x)

ddx=

(x)

eix eix

ddx

lm0

(x)eix (x)eix

dxd = lm0

(x)

eix eix

ddx

(x)

eix eix

d

0

(x)

eix eix

d

|

(x)

eix eix

d| = |

(x)

2i sin(x)

d| 2|(x)||

sin(x)

| 2|(x)|

x

T.C.D.

(x) lm0

eix eix

ddx

: R C(x) = lm

0

eix eix

d

x

x

R

|(x)| = | lm0

eix eix

d| = | < v p

1

, eix > | <


11/17

vp( 1

)

S(R)

eix

S(R)

< F(vp( 1x

)), >=

(x) lm0

eix eix

ddx=

F(vp( 1x

))(y) = (y) = lm0

eiy eiy

d

(y) = 2i lm0

sin(y)

d

u= y

sin(u)u

sin(u)

u 1[y,)

0

sin(u)

u 1[0,)

y >0

(y) =

2i lm0

y

sin(u)

u du=

2i

0

sin(u)

u du=

i

y


12/17

f

L1(Rd)

S

t > 0

S(t)f := 1

(4it)d/2

Rd

e|xy|2

4it f(y)dy

S(t)

L1

L

S(t)f

L

f

L1

),

||S(t)f||L C

td/2||f||L1

f

L1

|S(t)f| = | 1(4it)d/2

Rd

e|xy|2

4it f(y)dy| 1|(4i)d/2|td/2Rd

|e |xy|2

4it ||f(y)|dy

|eai| 1

|S(t)f| Ctd/2

Rd

|f(y)|dy = Ctd/2

||f||L1

||S(t)f||L C

td/2||f||L1

C= 1|(4i)d/2|

f L2(Rd) S(t)f

L2(Rd) S(t) L2

||S(t)f||L2 = ||f||L2

L1 L2

L2

L1

L2

L1

SL2 (t)f f L1 L2 SL2 (t)f =S(t)f f L2 (L1)c

fn L1 L2 fn f. S(t)f L1

SL2(t)f = L2 lmn

S(t)fn L2 L2

S(t)

L1

fn(x) =f(x)1[0,n](|x|)

L1

fn(x) f(x) f L2

SL2(t)f =L2 lm

n

1

(4it)d/2

Rd

e|xy|2

4it f(y)1[0,n](|y|)dy= L2 lmn

1

(4it)d/2

B(0,n)

e|xy|2

4it f(y)dy

L2.

SL2(t)

S(t)

L1 L2 L2 f

L1 L2

S(t)f= 1

(2)d2

Rd

eit|z|2

eizxf(z)dz

F(ea|x|2 )() = 1(2a)

d2

e||2

4a

|x y| = |y x|

S(t)f = 1

(4it)d/2

Rd

e|yx|2

4it f(y)dy = 1

(2)d2

Rd

F(eit|w|2 )(y x)f(y)dy


13/17

t > 0

1

(2)d2

Rd

Rd

1

(2)d2

eit|z|2

eiz(yx)dzf(y)dy = 1

(2)d

Rd

Rd

eit|z|2

eiz(yx)f(y)dzdy

f

|eit|z|2eiz(yx)f(y)| = |eit|z|2eiz(yx

z

eit|z|2

eiz(yx)

F(eit|w|2)(y x)

|f(y)| Rd

|eit|z|2eiz(yx)|dz f(y)

L

1

(R

2d

)

1

(2)d

Rd

Rd

eit|z|2

eiz(yx)f(y)dzdy= 1

(2)d

Rd

eit|z|2Rd

eiz(yx)f(y)dydz

1

(2)d

Rd

eit|z|2

eizxRd

eizyf(y)dydz= 1

(2)d2

Rd

eit|z|2

eizxf(z)dz

S(t)f= 1

(2)d2

Rd

eit|z|2

eizxf(z)dz

> 0

I = Rde2w2

4

|S(t)f

|2(w)dw = Rde

2w2

4 S(t)f(w)S(t)f(w)dw

|S(t)f|2

lm0

I =

Rd

lm0

e2w2

4 S(t)f(w)S(t)f(w)dw Rd

S(t)f(w)S(t)f(w)dw=

|S(t)f|2(w)dw= ||S(t)f||L2

S(t)f

I = 1

(2)d

w

e2w2

4

x

eit|x|2

eixwf(x)dx

y

eit|y|2

eiywf(y)dy

dw

S(t)f

f

L1

x

y

I= 1

(2)d

w

x,y

e2w2

4 eit|x|2

eixwf(x)eit|y|2

eiyw f(y)dxdydw

|e2w2

4 eit|x|2

eixwf(x)eit|y|2

eiyw f(y)| |e2w2

4 ||f(x)f(y)| L1

x

y ||f||L2 = ||f||L2 e2w24

Rd

I =

x,y

1

(2)deit|x|

2

eit|y|2

f(x)f(y)

w

e2w2

4 ei(xy)wdwdxdy =

x,y

eit|x|2

eit|y|2

f(x)f(y)G(x y)dxdy

G(z) = 1(2)d Rd e

izwe2|w|2

4 dw

eit|x|2

f(x)

y

eit|y|2

f(y)G(x y)dy0

eit|x|2

f(x)eit|x|2

f(x)

|eit|x|2 f(x)y

eit|y|2

f(y)G(x y)dy| |eit|x|2 f(x)|||f(y)||y

|G(x y)|dy |eit|x|2 f(x)|||f||K

G f L1 f L ||f||


14/17

L2

lm0

I = ||S(t)f||L2

lm0

I = ||f||L2

||S(t)f||L2 = ||f||L2 f L1 L2 S(t)f L2

S(t)f f Lp(Rd) 1 p 2 Lp Lp

||S(t)f||Lp C

t||f||Lp

[0, d2

]

S(t)

L1

L2

p0 = 1, q0 =, p1 = 2, q1 = 2 Lp 0 1 1p

= 1 + 2

= 1 2

Lq

1q

= 2

p

1p

= 1 1q

1p

+ 1q

= 1

q = p

p

0

1

1p

12

1

p

1

2

p [1, 2]

S(t)

L1 L

L2 L2.

S(t)

S(t)

L1 +L2 L +L2

f

L1

||S(t)f||L Ctd/2 ||f||L1 L2

g

L2

||S(t)g||L2 ||g||L2

M0 = Ctd/2

M1 = 1 S(t) Lp

Lp

||S(t)f||Lp M10 M1 ||f||Lp =

Ctd/2

1||f||Lp =

Ctd2(1)

||f||Lp =C

t||f||Lp

= d2

(1 )

0

1

[0, d2

]

C

C = 1

|(4i)d/2|

1

f D(Rd) S(t)f C(Rd) D(Rd)

S(t)f

f : Rn R R

x Rn f(x, ) C1

t R f(, t) L1

g L1 x t| ft

(x, t)| g(x)

F(t) :=Rn

f(x, t)dx C1

tF(t) =

Rn

f

t(x, t)dx


15/17

f

D(Rd)

S(t)f

x

Rd

xn x

lmn

S(t)f(xn) = 1

(4it)d/2 lmn

Rd

e|xny|

2

4it f(y)dy

e|xny|

2

4it f(y) n

e|xy|2

4it f(y)

|e|xny|

2

4it f(y)| |f(y)| L

1

f(y)

D(Rd)

lmn

S(t)f(xn) = 1

(4it)d/2

Rd

lmn

e|xny|

2

4it f(y)dy= 1

(4it)d/2

Rd

e|xy|2

4it f(y)dy= S(t)f(x)

S(t)f

xiS(t)f =

xi

1

(4it)d/2

Rd

e|xy|2

4it f(y)dy = 1

(4it)d/2

xi

Rd

e|xy|2

4it f(y)dy

g(x, y) = e|xy|2

4it f(y)

xi

g

C1

t > 0

xi L1

f

D(Rd)

|e |xy|2

4it | 1

|e |xy|2

4it f(y)| |f(y)| L1y

g(x,y)xi

= f(y)e|xy|2

4it xi

|xy|2

4it = f(y)e

|xy|2

4it 2(xiyi)

4it |xy |2 = dj=1(xj yj)2 |g(x,y)xi |

|f(y)2(xiyi)4it

| L1y x 2xi f(y)yi f

xi S(t)f =

1

(4it)d/2Rd

xi e|xy|2

4it

f(y)dy =

1

(4it)d/2Rd f(y)e

|xy|2

4it

2(xi

yi)

4it dy

xiS(t)f =

2xi4it

S(t)f+ 2

(4it)(4it)d/2

Rd

yif(y)e|xy|2

4it dy =2xi

4it S(t)f+

2

4itS(t)(f(w)wi)

xi

Rd

e|xy|2

4it f(y)dy

xi

S(t)f

S(t)f

f

D(Rd)

2xi4it

S(t)f

yif(y)

g(y)

g

D(Rd) S(t)g

S(t)f

C(Rd)

|xy|2 = |x|2+|y|2+2

S(t)f = 1

(4it)d/2

Rd

e|xy|2

4it f(y)dy= e

|x|2

4it

(4it)d/2

Rd

e|y|2

4it e2

4it f(y)dy= e

|x|2

4it

(4it)d/2

Rd

eie

|y|2

4it f(y)dy = e

|x|2

4it

(2it)d/2F(e |y|

2

4it f

S(t)f(x)(2it)d/2e|x|2

4it = F(e |y|2

4it f(y))(x

2t)


16/17

f

D(Rd)

e|y|2

4it f(y)

S(t)f(x)(2it)d/2e|x|2

4it

x2t

Rd

S(t)f

f S(Rd) S(t)f S(Rd)

u(t, x) := [S(t)f](x)

t >0.

f

S(Rd)

S(t)f S(R).

N, m

max||N

||(1 + |x|2)m|S(t)f|||<

m

|| <

||(1 + |x|2)m|S(t)f||| <

|| = 0 S(t)f

xi=

2xi4it

S(t)f+ 2

(4it)(4it)d/2

Rd

yif(y)e|xy|2

4it dy =2xi

4it S(t)f+

1

4itS(t)(f(w)wi)

f(w)wi f f || = 0 S(t)f

2xi4it

S(t)f+ S(t)(f(w)wi) = S(t)fxi

S(t)f

S(t)(f p)

p

f

S(t)f

|| = 0

||(1 + |x|2)m|S(t)f|||= || (1 + |x|2)m

|(4it)d/2| |Rd

e|xy|2

4it f(y)dy|||

1|(4it)d/2

|x y|2 = |y x|2 u+x = y

||(1 + |x|2)m| Rd

e|u|2

4it f(u+x)du|||= || Rd

(1 + |x|2)me|u|2

4it f(u+x)du||

||Rd

(1 + |x|2)m(1 + |x+u|2)m+p

(1 + |x+u|2)m+p e|u|2

4it f(u+x)du||

p

f |(1 + |x+u|2)m+pf(u+x)| Cm,p d m

x

||Rd

(1 + |x|2)m(1 + |x+u|2)m+p du|| <

S(t)f

S(t)f

f

S(t)f = 1

(2)d2

Rd

eit|z|2

eizxf(z)dz = F1(eit|z|2 f(z))(x)


17/17

F(S(t)f)(z) =eit|z|2 f(z)

t

t

tF(S(t)f)(z) =

t(eit|z|

2

)f(z) = i|z|2f(z)eit|z|2 = i|z|2F(S(t)f)(z)

t

F(S(t)f)(z) = i|z|2F(S(t)f)(z)

F1( t

F(S(t)f)(z)) = F1(i|z|2F(S(t)f)(z)) =iF1(|z|2F(S(t)f)(z))

f

f

|| 2

p

F(p(i)f)(z) =p(z)f(z)

p(z) = |z|2

p(ixi ) = ((ix1)2 + (ix2)2 +..+ (ixd)2 =x1 + ..+xd =

|z|2F(S(t)f)(z) = F(S(t)f)(z)

iF1(|z|2F(S(t)f)(z)) =iS(t)f

F1( t

F(S(t)f)(z)) = t

F1(F(S(t)f)(z)) = t

S(t)f

tF1(F(S(t)f)(z)) =

t

Rd

eixzeit|z|2

f(z)dz

eixzei|z|2

f(z)

C1

t

C

t

eixeit||2

f() eixF(S(t)f)() F1(F(S(t)f)(z))(x)

S(t)f

t

(eixzeit|z|2

f(z)) = i|z|2eixzeit|z|2 f(z) | i|z|2eixzeit|z|2 f(z)| =|eixz|z|2F(S(t)f)| L1 S(t)f |z|2F(S(t)f)

eixz|z|2F(S(t)f) F1(|z|2F(S(t)f)) F

|z|2F(S(t)f)

||z|2F(S(t)f)|

t

S(t)f = t

F1(F(S(t)f)(z)) = t

Rd

eixzeit|z|2

f(z)dz =Rd

eixz t

(eit|z|2

f(z))dz = F1( t

F(S(t)f)(z))

tS(t)f =iS(t)f

x

Rd

tS(t)f(x) = iS(t)f(x)

u(x, t) = S(t)f(x)

u(x, t)

t =iu(x, t)

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