Twisted Convolution, Peudo-differential Operators and Fourier Modulation Spaces
Transcript of Twisted Convolution, Peudo-differential Operators and Fourier Modulation Spaces
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arXiv:0811
.0519v2
[math.FA]12Dec2008
1 p 2 Lp()
Mp,q p, q [1, ]
Lp,q
p q
Mp,q
http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2http://arxiv.org/abs/0811.0519v2 -
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Mp,q() Wp,q()
Lp,q1,() Lp,q2,()
Wp,q()
Mp,q()
Wp,q()
M,1 M,1 L2
M,1 Mp,q p, q [1, ] M2,2 = L2
#
j pj, qj [1, ]
S(R2d) S(R2d) (a1, a2) a1#a2 S(R2d)
Mp1,q1(1) (R2d) Mp2,q2(2) (R
2d)
Mp0,q0(0) (R2d)
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C > 0
a1#a2Mp0,q0(0)
Ca1Mp1,q1(1)
a2Mp2,q2(2)
,
a1 Mp1,q1(1)
(R2d) a2 Mp2,q2(2)
(R2d)
j pj, qj [1, ]
S(R2d) S(R2d) (a1, a2) a1 a2 S(R2d)
Wp1,q1(1) (R2d) Wp2,q2(2) (R
2d)
Wp0,q0(0) (R2d)
C > 0
a1 a2Wp0,q0(
0)
Ca1Wp1,q1(
1)
a2Wp2,q2(
2)
,
a1 Wp1,q1(1)
(R2d) a2 Wp2,q2(2)
(R2d)
pj =
qj = 2 W2,2(j)
L2(j)
j j L2(1)(R
2d)
L2(2)(R2d) L2(0)(R
2d)
a1 a2L2(0)
Ca1L2(1)
a2L2(2)
,
a1 L2(1)
(R2d) a2 L2(2)
(R2d)
v R
d v
(x + y) C(x)v(y)
C x, y Rd v P(Rd)
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Rd
(x, ) P(R2d) x () (x) (x, ) P(R2d) P(Rd)
F
S(Rd)
(Ff)() = f() (2)d/2 Rd
f(x)eix, dx
f L1(Rd) F S(Rd) S(Rd) L2(Rd)
S(Rd) f S(Rd) Vf(x, ) f R2d
Vf(x, ) F(f ( x)().
f, S(Rd)
Vf(x, ) = (2)d/2
f(y)(y x)eiy, dy.
P(R2d) p, q [1, ] Lp,q1,()(R
2d) F L1loc(R
2d)
FLp,q1,()
< Lp,q2,()(R2d) F L1loc(R
2d)
FLp,q2,()
<
FLp,q1,()
=
|F(x, )(x, )|p dxq/p
d1/q
,
FLp,q2,()
=
|F(x, )(x, )|q dp/q
dx1/p
,
p = q = p, q [1, ] P(R2d) S(Rd) \ 0
Mp,q()(Rd)
f S(Rd)
fMp,q()
VfLp,q1,()
< .
Wp,q()(Rd)
f S(Rd)
fWp,q()
VfLp,q2,()
< .
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Mp,q()(Rd) Wp,q()(R
d)
Vbf(, x) = eix,Vf(x, ), (x) = (x),
f Wp,q()(Rd) f Mp,q(0)(Rd), 0(, x) = (x, ).
Mp() = Mp,p() W
p() = W
p,p()
Mp,q = Mp,q() Wp,q = Wp,q() 1
(x, ) = 1(x)2() 1, 2 P(Rd) Wp,q()
p [1, ] p [1, ] 1/p + 1/p = 1
p, q, pj, qj [1, ] j = 1, 2 , 1, 2, v P(R
2d) v 2 C1 C > 0
M1(v)(Rd)\0 f Mp,q()(R
d)
Mp,q()(Rd) Mp,q()
p1 p2 q1 q2
S(Rd) Mp1,q1(1) (Rn) Mp2,q2(2) (R
d) S(Rd)
L2 ( , ) S
Mp,q()(Rn) Mp
,q
(1/)(Rd) C a =
sup |(a, b)| b S(Rd) b
Mp,q
(1/)
1 Mp,q()
p, q < S(Rd) Mp,q()(Rd)
Mp,q()(Rd) Mp
,q
(1/)(Rd)
( , )L2 S(Rd) M()(R
d)
Mp,q() Wp,q()
M1(v) \ 0 C > 0
A S aWp,q()
C a A
M1(v) \ 0 a A M1(v) \ 0 C
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x = (1+|x|2)1/2 x Rd
p, p1, p2, q , q 1, q2 [1, ]
q1 min(p, p), q2 max(p, p
), p1 min(q, q), p2 max(q, q
),
, v P
(R
2d
)
v
p q Wp,q()(Rd) Mp,q()(R
d) p q
Mp,q()(Rd) Wp,q()(R
d) (x, ) = (x)
Mp,q1() (Rd) Wp,q1() (R
d) Lp()(Rd) Wp,q2() (R
d) Mp,q2() (Rd).
M2() = W2() = L
2() (x, ) = ()
Wp1,q() (Rd) Mp,q1() (R
d) FLq()(Rd) Mp2,q() (R
d) Wp2,q() (Rd).
FLq
(0)
(Rd) f S(Rd)
f 0Lq < ; (x, ) = (x)
Mp,q()(Rd) C(Rd)
Wp,q()(Rd) C(Rd)
q = 1
M1,(Rd) W1,(Rd) CB(Rd)
Rd
W1,(Rd) CB(Rd) M1,(Rd)
x0 Rd
0() = (x0, )
Mp,q() E = Wp,q() E
= FLq(0) E
(x, ) = 0(, x) S(Rd)
Mp()(Rd) Mp(0)(R
d)
= 0 Mp()
x, Rd
ei ,f( x)Mp,q()
Cv(x, )fMp,q()
,
ei ,f( x)Wp,q()
Cv(x, )fWp,q()
C f S(Rd)
(x, ) = (x, ) f Mp,q() f Mp,q()
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s R (x, ) = s M2() = W2() H
2s
s L2 H2s f S
F1( sf) L2
f S(Rd) P(R2d) S(Rd) p [1, ]
Hf,,p() =
Rd
|Vf(x, )(x, )|p dx
1/p.
f, fj Mp,q()(R
d) j = 1, 2, . . .
fj f p, q [1, ] S(Rd) \ 0 P(R2d)
fj f S(Rd) j
Hfj ,,p() Hf,,p() Lq(Rd) j
f, f1, f2, S(Rd)
Rd
liminfj
Hfj ,,p() Hf,,p() lim infj
fjMp,q()
fMp,q()
.
p, q [1, ] q < P(R2d) C0 (R
d) Mp,q()(Rd)
a S(R2d) t R at(x, D)
(at(x, D)f)(x) = (Opt(a)f)(x)
= (2)d
a((1 t)x + ty,)f(y)eixy, dyd.
S(Rd) a S(R2d) at(x, D)
S(Rd
)
S
(Rd
)
Kt,a(x, y) = (2)n/2(F12 a)((1 t)x + ty,y x),
F2F F(x, y) S(R2d)
y F2 F(x, y) F((1 t)x + ty,y x) S(R2d) a
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S(R2d) t = 0 at(x, D) a(x, D) t = 1/2 at(x, D) aw(x, D) a a, b S(R2d) a#b a b (a#b)w(x, D) = aw(x, D)bw(x, D)
R2d
(X, Y) =
(x, ); (y, )
= y, x,
, Rd a S(R2d)
(Fa)(X) = a(X) = d
a(Y)e2i(X,Y) dY .
F1 = F S(R2d)
S(R2d) L2(R2d)
a S
(R
2d
)
S(R2d)
Va(X, Y) = F
a ( X)
(Y), X, Y R2d.
P(R4d) Mp,q()(R2d) Wp,q()(R
2d)
Mp,q()(R2d) Wp,q()(R
2d) Mp,q()(R2d) Wp,q()(R
2d)
VF(Fa)(X, Y) = e2i(Y,X)Va(Y, X),
FMp,q()(R
2d) = Wq,p(0)(R2d), 0(X, Y) = (Y, X).
a, b S(R2d) a b
(a b)(X) = (2/)d/2
a(X Y)b(Y)e2i(X,Y) dY .
Lp(R2d) p [1, 2] S(R2d)S(R2d) S(R2d) a, b S(R2d)
a#b a b a#b = (2)d/2a (Fb).
F(a b) = (Fa) b = a (Fb),
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a(X) = a(X)
F(a#b) = (2)d/2(Fa) (Fb).
Wp,q()
a1 S(R2d) a2 S(R2d) 1, 2 S(R2d)
= d1#2 S(R2d)
Z e2i(Z,Y)(V1a1)(X Y + Z, Z) (V2a2)(X+ Z, Y Z)
L1(R2d)
V(a1#a2)(X, Y)
=
e2i(Z,Y)(V1a1)(X Y + Z, Z) (V2a2)(X+ Z, Y Z) dZ
= 2d1 2 S(R2d)
Z e2i(X,ZY)(V1a1)(X Y + Z, Z) (V2a2)(Y Z, X+ Z)
L1(R2d)
V(a1 a2)(X, Y)
=
e2i(X,ZY)(V1a1)(X Y + Z, Z) (V2a2)(Y Z, X+ Z) dZ
0(X, Y) C1(XY+Z, Z)2(X+Z, YZ), X, Y, Z R2d.
C > 0 pj, qj [1, ]
1
p1+
1
p2
1
p0= 1
1q1
+1
q2
1
q0
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0 1
p1+
1
p2
1
p0
1
pj,
1
qj
1
q1+
1
q2
1
q0, j = 0, 1, 2.
0, 1, 2 P(R4d)
pj, qj [1, ]
j = 0, 1, 2
S(R2d) Mp1,q1(1) (R
2d) Mp2,q2(2) (R2d) Mp0,q0(0) (R
2d) C >
0 a1 Mp1,q1(1)
(R2d) a2 Mp2,q2(2)
(R2d)
0(X, Y) C1(XY+Z, Z)2(YZ, X+Z), X, Y, Z R2d.
0
1
q1 +
1
q2
1
q0
1
pj ,
1
qj
1
p1 +
1
p2
1
p0 , j = 0, 1, 2.
0, 1, 2 P(R4d)
pj, qj [1, ] j = 0, 1, 2 S(R2d) Wp1,q1(1) (R
2d) Wp2,q2(2) (R2d) Wp0,q0(0) (R
2d) C >
0 a1 Wp1,q1(1)
(R2d) a2 Wp2,q2(2)
(R2d)
0, 1, 2 P(R2d) p, p1, p2 [1, ]
0(X1 + X2) C1(X1)2(X2), p1, p2 p
max1
p,
1
p
1
p1+
1
p2
1
p 1,
C Lp1(1)(R
2d)Lp2(2)(R2d) Lp(0)(R
2d) C
a1 a2Lp(0) Ca1Lp1(1)
a2Lp2(2)
,
a1 Lp1(1)
(R2d), a2 Lp2(2)
(R2d).
p1 p2
p2 <
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W2() = M2() = L
2() (X, Y) = (X)
p1 =p2 = p = 2
1/p1 + 1/p2 1/p = 1 a1 Lp1(R2d)
a2 S(R2d)
a1 a2Lp(0) (2/)d/2
|a1| |a2| Lp(0) Ca1Lp1(1)a2Lp2(2),
S Lp2(2) p2 <
p1 = p2 = p = 2 1/p1 + 1/p2 1/p =1
(Lp1()(R2d), (Lp2()(R
2d))[] = Lp0()(R
2d),
1
p1+
p2=
1
p0.
p1 = p p2 = q min(p, p) p2 = p p1 = q
min(p, p)
0, 1, 2 P(R2d) p, q [1, ]
0(X1 + X2) C1(X1)2(X2), q min(p, p)
C Lp(1)(R
2d) Lq(2)(R2d) Lq(1)(R
2d) Lp(2)(R2d)
L
p
(0)(R
2d
)
C
a1 a2Lp(0)
Ca1Lp(1)
a2Lq(2)
, a1 Lp(1)
(R2d), a2 Lq(2)
(R2d)
a1 a2Lp(0)
Ca1Lq(1)
a2Lp(2)
, a1 Lq(1)
(R2d), a2 Lp(2)
(R2d).
k {1, 2} 0, 1, 2 P(R
2d) p, pj, q , q j [1, ] j = 1, 2
0(X1 + X2) C1(X1)2(X2), p1, p2 p, q1, q2 q,
max1
p,
1
p,
1
q,
1
q
1
p1+
1
p2
1
p 1
max1
p,
1
p,
1
q,
1
q
1
q1+
1
q2
1
q 1,
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C Lp1,q1k,(1)(R
2d) Lp2,q2k,(2)(R2d) Lp,qk,(0)(R
2d) C
a1 a2Lp,qk,(0)
Ca1Lp1,q1k,(1)
a2Lp2,q2k,(2)
,
a1 Lp1,q1k,(1)(R2d), a2 Lp2,q2k,(2)(R2d).
p1 = q1 = 1 p2 = q2 = 1
p1 = p2 = q1 = q2 = 2
p, p0, q , q 0 [1, ] , v P(R2d)
p0, q0 min(p, p, q , q ) v = v v
f S(Rd)
Mp0,q0(v) (Rd) \ 0 f Mp,q()(R
d) Vf
Lp,q1,()(R2d) f VfLp,q
1,()
Mp,q()(Rd)
Wp0,q0(v) (Rd) \ 0 f Wp,q()(R
d) Vf
Lp,q2,()(R2d) f VfLp,q
2,()
Wp,q()(Rd)
f S(Rd) g S(Rd)
Wf,g(x, ) = F(f(x/2 )g(x/2 + )().
f, g S(Rd) Wf,g
Wf,g(x, ) = (2)d/2
f(x y/2)g(x + y/2)eiy, dy.
Wf,g(x, ) = 2deix,/2Vgf(2x, 2),
Wf,Lp,qk,(0)
= 2dVfLp,qk,()
, 0(x, ) = (2x, 2)
k = 1, 2
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f1, g2 S(Rd) f1, g2 L
2(Rd)
Wf1,g1 Wf2,g2 = (f2, g1)L2Wf1,g2.
p0 = q0 = min(p, p, q , q )
, Mp0,q0(v) (Rd) L2(Rd)
p0, q0 2 v c c > 0 v v VLp0,q0
k,(v)= VLp0,q0
k,(v)
v = v
VfLp,qk,()
CVfLp,qk,()
VLp0,q0k,(v)
,
C f S(Rd) , Mp0,q0(v) (R
d)
p1 = p p2 = p0 q1 = q q2 = q0 0 = (2 ) v0 = v(2 )
VfLp,qk,()
= C1Wf,Lp,qk,(0)
= C2Wf, W,Lp,qk,(0) C3Wf,Lp,qk,(0)
W,Lp0,q0k,(v0)
= C4VfLp,qk,()
VLp0,q0k,(v)
,
Wp,q()
p, q [1, ] 1, 2 P(R2d)
P(R4d)
2(x, + )
1(x + y, ) C(x, , , y)
C x, y, , Rd a Wq,p()(R
2d) a(x, D) S(Rd)
S
(Rd
)
Mp,q
(1) (Rd
) Wq,p(2)(R
d)
a(x, D)fWq,p(2)
CaWq,p()
fMp
,q
(1)
C f Mp,q
(1)(Rd) a
Wq,p()(R2d)
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a S(R2d) f, g S(Rd) T
(T )(x, ) = (, x)
S(R2d)
(a(x, D)f, g)L2(Rd) = (2)d/2(T(Fa), Vfg)L2(R2d),
f1, f2 S(Rd) p, q [1, ] 0
P(R4d) 1, 2 P(R2d) 1, 2 S(R
d) = V12
0(x, , , y) C1(x y, )2(y, + ) C
V(Vf1f2)Lp,q1,(0) CV1f1Lp,q1,(1)
V2f2Lq,p2,(2)
1(x y, )2(y, + ) C0(x, , , y)
C
V1f1Lp,q1,(1)V2f2Lq,p2,(2)
CV(Vf1f2)Lp,q1,(0)
C f1 Mp,q(1)
(Rd)
f2 Wq,p(2)
(Rd) Vf1f2 Mp,q(0)
(R2d)
C1Vf1f2Mp,q(0) f1Mp,q
(1)f2Wq,p
(2) CVf1f2Mp,q(0)
,
C f1 f2
p < q < p = q =
|Vf1(x y, )Vf2(y, + )| = |V(Vf1f2)(x, , , y)|
F1(x, ) = V1f1(x, )1(x, ) F2(x, ) = V2f2(x, )2(x, ),
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V(Vf1f2)qLp,q1,(0)
=
R2d
R2d
|V(Vf1f2)(x, , , y)0(x, , , y)|p dxd
q/p
dyd
CqR2d
R2d
|F1(x y, )F2(y, + ))|p dxd
q/pdyd.
y + x y
V(Vf1f2)qLp,q1,(0)
Cq
Rd
Rd
|F1(x, )|p dx
q/pd
Rd
Rd
|F2(y, )|p d
q/pdy
= CqV1f1qLp,q1,(1)
V2f2qLq,p2,(2)
.
C = 1 1 < p 1 < q
0(x, , , y) = (y,,, x)1,
a Wq,p()(R2d)
f, g S(Rd) a(x, D)f
S
(R
d
)
VfgMp,q(0)
CfMp
,q
(1)
gWq
,p
(12 )
.
T
(V(Ta))(x, , , y) = ei(x,+y,)(VTba)(y,,, x).
|(V(Ta))(x, , , y)0(x, , , y)1|= |(V1a)(y,,, x)(y,,, x)|,
1 = T Lp,q1 TaMp,q(10 )
=
aWq,p()
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|(a(x, D)f, g)| = (2)d/2|(Ta, Vgf)| C1TaMp,q
(10 )
VfgMp,q(0)
C2aWq,p()fMp,q(1) gWq,p
(12 ).
S(Rd) Mp,q
(1)(Rd)
Wq,p
(12 ) Wq,p(2) p, q > 1
p = 1 q < q = 1 p < f Mp
,q
(1) a S(R2d) a(x, D)f
S(Rd)
S(R2d) Wq,p()(R2d) p q
p = q = 1 p = q = S
M,1() W1,()
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7/30/2019 Twisted Convolution, Peudo-differential Operators and Fourier Modulation Spaces
17/18
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7/30/2019 Twisted Convolution, Peudo-differential Operators and Fourier Modulation Spaces
18/18