twisted convolution, peudo-differential operators and fourier modulation spaces

7/30/2019 Twisted Convolution, Peudo-differential Operators and Fourier Modulation Spaces

1/18

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


2/18

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)


3/18

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)


4/18

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,()

< .


5/18

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


6/18

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()


7/18

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


8/18

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),


9/18

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


10/18

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 <


11/18

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,


12/18

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


13/18

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)


14/18

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, ),


16/18

|(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,()


17/18


18/18

twisted convolution, peudo-differential operators and fourier modulation spaces

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