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D(R) S(R)
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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
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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
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< 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
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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)|
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|(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
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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|
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x
n
max||N
sup|x|
lmn
< Tn, >< ex, >
Tn ex
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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
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| 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 > | <
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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
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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
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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||
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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
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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)
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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)
7/25/2019 Tarea1versionfinal1
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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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