Scholarly article on topic ' Remarks on the Unimodular Fourier Multipliers on α -Modulation Spaces '

Remarks on the Unimodular Fourier Multipliers on α -Modulation Spaces Academic research paper on "Mathematics"

CC BY
0
0
Share paper
Academic journal
Journal of Function Spaces
OECD Field of science
Keywords
{""}

Academic research paper on topic " Remarks on the Unimodular Fourier Multipliers on α -Modulation Spaces "

Hindawi Publishing Corporation Journal of Function Spaces Volume 2014, Article ID 106267, 8 pages http://dx.doi.org/10.1155/2014/106267

Research Article

Remarks on the Unimodular Fourier Multipliers on a-Modulation Spaces

Guoping Zhao,1 Jiecheng Chen,2 and Weichao Guo3

1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China

2 Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China

3 Department of Mathematics, Xiamen University, Xiamen 361005, China

Correspondence should be addressed to Weichao Guo; weichaoguomath@gmail.com Received 15 April 2014; Revised 28 June 2014; Accepted 28 June 2014; Published 17 July 2014 Academic Editor: Yuri Latushkin

Copyright © 2014 Guoping Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the boundedness properties of the Fourier multiplier operator on a-modulation spaces M^ (0 < a < 1) and Besov spaces B^M*'^). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness of Mspaq. If is a radial function $(№) and $ satisfies some size condition, we obtain the sufficient and

necessary conditions for the boundedness of between Mi1and MSJ?'" .

1. Introduction

Let F and F-1 denote the Fourier transform and the inverse Fourier transform, respectively. For a bounded function m, the Fourier multiplier operator associated with m is defined by

Tm (f) (x) = m(D)f = F-1 (mFf)

on all Schwartz functions f e S(R"), where m is called the symbol or multiplier of Tm. Fourier multipliers arise naturally from the formal solution of linear partial differential equations and from the summabilities of Fourier series. The boundedness properties of a Fourier multiplier in various function or distribution spaces contribute an important research topic in harmonic analysis, as well as many significant applications in partial differential equations.

Let X and Y be two function/distribution spaces with

norms (or quasinorm)

f, respectively. A

bounded function m is called a Fourier multiplier from X to Y, if there exists a constant C > 0 such that

\K (f)\\r *C\\f\\X' (2)

for all f e S(R"). We use the above definition to avoid the situation where S(R") is not dense in MS'1 when p = œ> or q = <m.

In this paper, we will study the unimodular Fourier multipliers on the a-modulation space M^ (a e [0,1]) (see Section 2 for the definition of M^). Particularly, we will focus on the unimodular Fourier multipliers with symbol for real-valued functions These multipliers arise when one solves the Cauchy problem for some dispersive equations. For example, for the Cauchy problem of (linear) Klein-Gordon equations

utt + (I - A) u = 0, u (0, x) = u0, ut (0, x) = u1, (t, x) e R x R", the formal solution is given by u (t, x) = K' (t)u0 + K (t) u1,

G(t)-G(-t)

K(t) =

2i(l + \D\

and the Klein-Gordon semigroups are defined by

G{t)=e>t(1 + \D\2)m.

The modulation spaces were introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. Now, people have recognized that the modulation spaces are very important function spaces, since they play more and more significant roles not only in harmonic analysis, but also in the study of partial differential equations. On the other hand, Besov space Bsp is also a popular working frame in harmonic analysis and partial differential equations. In 1992, Grobner introduced the a-modulation space Ms'a„ [2], that is an intermediate space between these two types of spaces with respect to the parameters a e [0,1]. Modulation spaces are special a-modulation spaces in the case a = 0, and the (inho-mogeneous) Besov space Bsp can be regarded as the limit case of Ms'an as a ^ 1 (see [2]). So, for the sake of convenience, we can view the Besov spaces as special a-modulation spaces and use Msp1 to denote the inhomogeneous Besov space Bsp .

It is known that is not bounded on any Lebesgue space Lp and Besov spaces Bsp , except for p = 2 or ft = 1

and n = 1, (see [3, 4]). However,

bounded on the

modulation space Msp = Msp0q for all 1 < p,q < ot, s e R (see Benyi et al. [5]). Hence, the modulation spaces play an alternative role in the study of unimodular Fourier multipliers. In [5], the authors proved that if 0 < p < 2,

bounded on Ms for all 1 < p,q < ot, s e R. Furthermore, in the case p > 2, Miyachi et al. [6] showed that, for 1 < p,q < ot and s^ s2 e R, e'IDI is bounded from Mp,q to Msp]q if and only if ^ - s2 > (¡3-2)n\1/p- 1/2|.The reader also can see [7-11] for more results in this topic.

Since the «-modulation space Ms'an is an extension of the

classical modulation space and it is a natural bridge connecting the modulation spaces and the Besov spaces (see [12,13]), in a recent paper [14], we study the boundedness of

e'^ on

function spaces Mspaq and establish a sufficient and necessary boundedness theorem by assuming that ^ is a homoge-nous function. Thus, it will be interesting to study when ^ is not a homogenous function. This motivates us to seek some sharp condition to ensure the boundedness on Ms'a„ for

the unimodular multiplier ewhen ^ is not homogenous. In this note, we will focus on the case that ^ is a radial but not homogeneous function. We remark that, for a radial function the operator enot only is a generation for the Schrodinger semigroup e'IDI , but also works for the KleinGordon semigroup with symbol ev,where = (1 + |£|2)1/2 is not homogeneous.

We now present our main results.

Theorem 1. Let S > 0, Le N, L> [n/2] + 1, p > 0, and SP = SP (?) = (j-l) max {(p-2)n + 2an, 0}. (7)

Assume that ^ is a real-valued function of class CN (N > L, [n/2] + 3) on R" \ {0} which satisfies

y\ ß-M

0 < \ * \ < 1, > 1, 2<\v\

Suppose also that 1 < p,q < ot, st e R, a e [0,1],for i = 1,2, and satisfies ^ - s2 > |Sp|. Then one has

<c\\f\\Milf,

where the constant C is independent of f.

Corollary 2. Let S > 0, L e N, L> [n/2] + 1, 9 > 0, X > 0, and

S'p (p) = (1- 1) max {(p-2)n + 2an,0}. (10)

Assume that $ is a real-valued CN(R \ {0}) (N > L, [n/2] + 3) function satisfying

\dk$(r)\<Ck\rf-k, 0<\r\<1,k = L,

\dk$(r)\ < Ck\r\6-]k], \r\ > 1, 1<\k\<

Let h: R" \ {0} ^ R \ {0} be a smooth positive homogeneous function with degree X. Suppose 1 < p,q < ot, st e R, a e [0,1] for i = 1,2, andsatisfies s1 -s2 > |Sp(A0)|. Thenonehas

II \ I ¿Vi-p^ p,q

where the constant C is independent of f.

Theorem 3. Let 8 > 0, L e N, L> [n/2] + 1, ¡3 > 0, ¡3=1, X > 0, and

Sp = (P) = (1-1) max i(p -2)n + 2an, 0}. (13)

Assume that $ is a real-valued CN(R+) (N > L, [n/2] + 3) function which satisfies

\dk$(r)\<Ck\rf-k, 0<\r\<1,k = L

\dk$(r)\ < Ck\r\Hk], \r\ > 1, 1<\k\<

(14) + 3, (15)

\дk$(r)\~\rf-k, ^>1^=1,2. Let 1 < pt,qt < ot, st e R, a e [0,1] for i = 1,2. Then

¥mi)f\L- * Mm:-

P2>12 PI'11

holds for all f ifand only if

P2 < Pi'

52 -1T+ maX ' SP[} = -

q2 < qi

1 < 1 ft " Pi '

{sft -S {^2 - S

' } < ^ -

n (1 - a) <?2

na n (1 - a) < s,--+ —--.

We list two examples to illustrate the assumptions in our

theorems. First, the function = (1 + A;-|£| j) (A;- > 0, A > 0,0 > 0) satisfies the assumptions in Theorem 1 and Corollary 2 for ^ = A0 > 0, while is not radial and not homogeneous. Another function is = ^(|£|) with ^(r) = (1 + rA)e (A,0 > 0). This function satisfies the assumptions in Theorem 3 for ^ = A0, A0 = 1. One may also observe that if ^ = 1, there exists no C2(R+) function $(r), which satisfies the size condition (16). If the reader checks the main theorems in [5, 6], it is not difficult to see that our theorems are a substantial improvement and extension of the known results, even in the case a = 0.

The paper is organized as follows. In Section 2, we recall some definitions and basic properties. In Section 3, we obtain an improvement of results in [5, 6] by studying more general Fourier multipliers e^(D) ,in which we do not need to assume lower order derivatives of near 0. This new results will be used to achieve a more general result for the boundedness of ev(D) on spaces M*'^. In Section 4, by assuming radial

condition on we deduce a dual estimate of e"^(|D|), and then we use the method in [14] to give a sharp result for the boundedness of ei0(|D|) between Ms''" and Ms„2'" .

2. Preliminaries

We start this section by recalling some notations. Let C be a positive constant that may depend on the indices n,pi,qi, , a, The notation X < Y denotes the statement that X < CY, the notation X ~ Y means the statement X < Y < X, and the notation X - Y denotes the statement X = CY. For a multi-index fc = (fc1,fc2,...,£„) e Z", we denote |fc|m := sup ^..Jfcl and <fc> := (1 + |fc|2)1/2.

Let S := S(R") be the Schwartz space and S := S'(R") the space of all tempered distributions. We define the Fourier transform Ff and the inverse

Fourier F-1/ of f e S(R")

F/(Ç) = /(Ç)=f /(*)<

F-1/(x) = /(-*) = f /(fle2**^.

To describe the function spaces discussed in this note, we first give the partition of unity on frequency space for

a e [0,1). We suppose c > 0 and C > 0 are two appropriate constants and choose a Schwartz function sequence {^}fc6Z» satisfying

\ a/(1-a)_

> 0, if |^-fc<fcr/(1-a)| <c(fc>

supp ^ C : fc<fc>a/(1-a)| < C<fc>a/(1-a) j

V^e R";

(^)|<C|a|<fc>-a|^|/(1-a); e R", ye (Z+ u {0})".

Then {^*(£)}/c6Z» constitutes a smooth decomposition of unity. The frequency decomposition operators associated with above function sequence can be defined by

□fc := F-1%F

for fc e Z". Let 1 < < ot, s e R, and a e [0,1); the a-modulation space associated with above decomposition is defined by

M- (R") =

f e S' (R")

hm^(r»)

= ( I <*)

s?/(1-a)|| a

□Ê/II

with the usual modifications when q = œ>. For the sake of

is = Ms,°

simplicity, in this note, we always denote M* = Mp° and

We introduce the dyadic decomposition of R" in order to define the Besov space. Let <p(£) be a smooth bump function supported in the ball {£ : |£| < 3/2} and be identically equal to 1 on the ball {£ : |£| < 4/3}. We denote

and a function sequence

^ = je N

For all integers j e N, we define the Littlewood-Paley operators

Â0? = y(Ç)/(Ç).

Let 1 < _p, ^ < œ, and se R. For f e S' we set the the (inhomogeneous) Besov space space norm by

= ( I2j1|Ai/||

The (inhomogeneous) Besov space is the space of all tempered distributions / for which the quantity ||/||Bs is finite. We recall the following embedding results.

Lemma 4 (embedding [12,13]). Let 1 < < to, e R, for i =1,2, a e [0,1]. If

P2 " P1,

«a na s2--= si--,

2 P2 1 Pi

1 < 1 <?2 " <?1

P2 " Pi «a

Si--< s, -

na n (1 - a)

S2 - T" + -

P2 <?2

< Si--+

na n (1 - a)

one has

Pi <?i

Ms> c Msfl2 .

We also recall some results obtained in [6, 14], respectively.

Lemma 5 (see Lemma 3.2 in [6]). Let e > 0 and let ^ be a c[«/2]+i function on R" \ {0} satisfying

for |y| < [n/2] + 1. Then F-1[^eiM?)] e L1(R") for each q e C~(R").

Lemma 6 (see Theorem 1.1 in [14]). Let p > 0, e > 0, and S = S (0) = (1 - 2 ) max {(0 -2)« + 2an, 0}. (32)

Assume that ^ is a real-valued function of class c["/2]+3 on R" \ {0} which satisfies

|3V(Ç)|<Cy|Ç|' |3V(Ç)|<Cy|Ç|'

s—jyj

0<|Ç|<1, |y|< l^l>1, 2<|y|<

L 2 J n

Suppose also that 1 < _p, q < to, e R, a e [0,1],for« = 1,2, and they satisfy s1 - s2 > |Sp |. Then we have

IK^/IU < Cl/I^Jl.,

II n.iv.ip^, p,q

where the constant C is independent of f.

3. Sufficient Condition of the Boundedness of e>(D)

The goal of this section is to prove Theorem 1 and Corollary 2. We will start with the following derivative lemma for showing that the lower order derivative near 0 does not interrupt the boundedness of e^(D) on a -modulation spaces.

Lemma 7 (derivative lemma). Let L e Z, L > 0, 5 > 0. Suppose that m e Cl(R" \ {0}) satisfying

|9ym(0|<Cy|^ryl,

0 < IÇI < 1,

Then the limit

a = lim m (£) = m (0)

exists, and for any e e (0,1) n (0,5], we have

|9y (m-fl)(^)|<Cy|^|E-jyj, 0 < < 1,

< L. (37)

Proof. We will state the proof for the cases L = 0,1,2; the other cases can be deduced by a similar argument and an easy induction.

For L = 0, one can observe directly that a = 0 = lim?^Q rn(0.

For L = 1, fix e e (0,1) n (0,<5). For any £2 e B(0,1), 1 < |£2|, we can find a simple piecewise smooth curve

r : [0,/] R",

which is jointed by two curves r and r2 with length I = |r|, where r is the straight line connecting points ^ and (|£2|/ and r2 is shortest curve on the great circle connecting points (|£2|/|£i|) • and £2, such that

r(0) = ç1, r(|?2|-|?i|) = &i, r(/) = ?2.

I^il (39)

|r' (i)| = 1

for all t e [0, /]. We have

= |m(r(/))-m(r(0))|

<|m(r(/))-m(r(|^2|-|^i|))| + |m(r(|^2|-|^i|))-m(r(0))|, Mr(|^H^i|))-m(r(0))|

fj?2j-j?ll

(Vm) (r(i))-r' (i)

j?2j-j?lj

|(Vm) (r(i))|di

m(r(Z))-m(r(|^|-|^|))| = |f (Vm)(r(i))-r' (i)di

IJi?2i-i?ii

< f |(Vm)(r(i))|di

J|f,i-if,i

Hence,

hfe) -™< ^^ + l^r — o (41)

as ^ O.Sothelimit

exists and

a = limm (£) = m (0)

| m (£) - a| = lim |m (£) - m < + < |£|e.

For L = 2, fix e e (0,1) n (0,<5). For any £ e B(0,1), let r be the straight line connecting £ and £/|£|, such that

r (0) = r(i-|?|) = || |r' (i)| = 1.

For any j = 1,2,..., n,we have

9,-m (£) - 9,-m

= |(9jm) (r(l-|^|))-(ajm}(r(0))|

= |jHi'(Vajrn)(r(i))-r/ (i)di

f1-i?i|/ N I

<J |(vajm)(r(i))|di

<I (Kl+f)*

It follows that

9,-m (£) - 9,-m

äjmlfi|

<1^ + 1<|ir.

We are in a position to give the proof of Theorem 1. Proof of Theorem 1. In virtue of the above lemma, since

0 < | *| < 1, |y| = L, and L > [n/2] + 1, for fixed e e (0,1) n (0,5], we have

0 < kI < 1

| 9r (p-p(0))tf)|<Cy|*|e

for any |y| < [n/2] + 1.

Using Lemma 5, we know that

F-1 = F-1 e L1.

Finally, we use Lemma 6 to complete the proof.

(49) □

Proof of Corollary 2. By the assumptions of Corollary 2, we can use Lemma 7 to deduce that

|afc (0-0(0))(r)| <cfc |r

0 < |r| < 1

for fixed e e (0,1) n (0,5] and any k < [n/2] + 1. The assumption of implies that

It follows then

K(<^)(0|<|^Hr', 0 < |£| < 1, (52)

for any |y| = [n/2] + 1.

On the other hand, one can deduce that

(0o^)(^)|<|^|A0-M, |^|>1, (53)

for 2 < |y| < [n/2] +3.

Finally the conclusion is deduced by Theorem 1. □

4. Sharpness of the Conditions for the Boundenness of

In this section, we give the proof of Theorem 3. The key point is that we can obtain a dual estimate for e"^'® under some size condition on 0. By combining the dual estimate with Theorem 1, we get the simultaneous asymptotic estimates of ll/ll^s,,* and . Then the proofcanbefinishedbythe

method in [14]. We first start with the dual estimate on Besov spaces.

Lemma 8 (dual estimate for 5^). Suppose p > 0, 1.

Assume that 0 is a real-valued c["/2]+3 function which satisfies the assumptions of Theorem 3. Then one has

||F-1 [y, ©e

;0(i?i)i

j03-2)(-«/2)

Now we finish the proof by repeating the L = 1 case. □ for all j e N.

Proof. Using the change of variables, we have

f [vj me'^]^ = 2-'"\\f—1

Use the polar coordinates,

F-1 "](*)

= f f(Z)e^2' l 1 e2mx*dt,

f~ , f , (56)

= f v (r) e'^r"-1 J1 e2mrx* da (?) dr

= f f(r)ei^(2r)rn-1da(-rx)dr. Jo

Recall that the Fourier transform of the area measure satisfies

2nirlxl

do(-rx) = C

(n—1)/2

-2nir\x\ a("-1)/2

+ o(\rx\—n/2).

The support of f yields 1/\rx\(n-m < 1/\x\(n-1)l2 and 0(\rx\-nl2) < \x\-n'2.

For the case \x\ > 2$, we only need to show that

¡[(f>(2'r)±2nir\x\] n-1

n—1 1

I r œ

which is a direct conclusion by the fact that d2 (2jr)±2nir\x\)

-jß(n/2)

and the Van der Corput lemma.

For the case that \x\ < 2^, we define

?J=1 W,

i[2j & Kfl + Z^ 2nx^\

and notice that

L(eiW2>r)±2mrlxl] \ = ei[$(2>r)±2mrlxl]

Then the inequality

\f fWe^'^e^dt

I J R"

follows by an integration by parts. Hence, for all j e N,

(2jß)

—n/2

\\F—1 [fj (O^IU <2>(B—2)(—n/2).

(63) □

Lemma 9 (dual estimate for Mpaq, a e [0,1)). Suppose ¡3 > 0,

¡3=1. Assume that (f> is a real-valued c^nl2^+3 function satisfying the assumptions of Theorem 3. Then there exists a sufficiently large constant R such that

\\F-1 [4 (I;) e"fm]\\Lm < c(k)W-2)l(1-a))(-nl2), (64)

for all k e Zn with {k)al(1-a)k e R \ B(0,R), where the constant C is independent of k.

Proof. For sufficiently large k, we use estimate on some A j to

estimate ak. Choose a j satisfying {k)1l^1-a?l ~ An easy computation shows that

..« ¿m\) 1

\\f—1 [nak (fte™®

< \\-F~1 [yj (£)4 (t)e'

< IF-1 [yj (O^j „i00

< 2j(ß—2)(—n/2) ^ ^((ß—2)/(1—a))(—n/2)

Now, we give the asymptotic estimates of f\\Ms,«

and ||/\\Ms,«. These results can be verified by the same methods in [14].

Lemma 10 (asymptotic estimates of \\e'^Dl f\\Ms,« (a e [0, 1))). Suppose ¡3 > 0, (3=1. Assume that (f> is a real-valued function which satisfies the assumptions of Theorem 3. we can find function sequences [fx], [hk], and such that

№D\)

fx\\ - ~Än(1—(1/P)) asX-^0,

J A\\Mv >

e'mX\\Ms,« ~ {k)(3+S?+an(1—1/Pm1—a) as \k\ to,

\y<P(D\)a \\ ^ N((3+Sp+an(1—1/p))/(1—a))+(n/q) \\ yN\\Msl;aq ~

na n(l - a)

as N —> to, s---+ S„ +--> -na,

P^VnL,« ~(lnNfi asN-^TO,

na n(l - a) s--+ SD +-= -na.

Moreover, we can easily verify that

№\mz ^n(1-{1lp))

\\hk\\M„ - {k)(^<1-1le))l(1-«) as \k\

1NÏÏM:

^ ^ N((3+an(1—1/p)/(1—a))+(n/q) asN_^TO,

na n(l - a)

s---1--> -na,

~(ln N)1/q asN-^œ,

na n(l - a)

s---1--= -na.

Proof. In the case that 0 < p < A a is trivial, we suppose that p > A a in this proof. We only show the proof of (66).

Denote by h(%) a C™(R") function with sufficiently small support near zero. Let

fx = h

Using Theorem 1, we deduce that

\\MM;:> u < ¥mn ML^ \\a

Notice that

wmm^ = wmm^\a = hm»

= Xn\\h(Xx)\\LP =\«1-(1'P»\\h\y;

so we have

\\emDl)fA .a ~ \n(1-(Up)) as X —> 0.

J A \ \ Ms,a

For the second equation, we denote

't-ikr'^k'

hk (0 = h

a/(1-a)

and let fk($) = hk(Q, where k e Z" with (k)al(1-a)k e r \ B(0,R).

If 1 < p < 2,we use Theorem 1 and Lemma 9 to deduce

\\fkll^+a/i-i/w-2)».« < \ \fk\ \< \\/fc\\Ms+sP,a. (73) If 2 < p < >x>, we deduce that

WAWm'+'V < \\e,mnML° < \\fk\Ls+(1/f,-1/2W-2)""*. (74)

By a direct calculation, we obtain

Ukl^'P" = (k)(s+Swl-a)\\hk\\„

= {k)is+Sp+Knil-llp))lil-K)\\h\y, Ms+(1/p-1/2)(^-2)».» (75)

= (lk)(s+SP+an(1-llP))l(1-a) \\h\\ . Then the asymptotic formula follows.

For the third and fourth asymptotic formulas, let

in (Z)=l\ (H), (76)

where AN = [k e Zn : \k\ < N, {k)a'{l-a)k e Rn \ B(0,R)}. By a direct calculation, we obtain

\\fN\\^ ^f £ (^s+^+an^Upmi^ ) \kiAN /

~ \\/n\\ms+(1/p-1/2)(^-2)"'" .

As above, Theorem 1 and Lemma 9 yield

Nll M!,-01

((s+Sp+an(1-1/p))/(1-a))q

\ k£AK

Finally, the claim follows by

y k^A^

((s+Sp+an(1-1/p))/(1-a))q

„ N((s+Sp+œn(1-1/p))/(1-a))+(n/q)

for 5 - (na/p) + Sp + (n(\ - a)/q) > -na, and

( Y (k)((s+S^+an(1-1/P))/(1-a))q) ~(lnN)1/q (80)

\knAN /

for 5 - (na/p) + Sp + (n(\ - a)/q) = -na.

For the case a = 1,we have the following lemma; since its proof is similar to that of Lemma 10, we leave the detail to the reader.

Lemma 11 (asymptotic estimates

p > 0, p=1. Assume that $ is a real-valued function which satisfies the assumptions of Theorem 3. We can find function sequences [fx], [hk], and |^N| such that

J^(\D\)

fxh ~^n(1-(1/P)) osX-^Q,

¥m\L ~ 2ils+S?+n(1-1/P)] as \j\ -^œ,

2Nls+Sp +n(1-1/P)] as N

s--+ Sp > -n,

Ik^WL asN-^œ,

s--+ S p = -n.

Moreover, we can easily verify that

ll/J, -A«™' as A

11 }llBU

.2N[s+n(1-1/P)] asN-

s--> -n,

7n|ib?

s--= -n.

Now, we give the sketch of proof for Theorem 3. Proof of Theorem 3. As in [14], if

,>0(\D\)

holds for all /, we can use Lemmas 10 and 11 to obtain (18) or (19).

On the other hand, if (18) or (19) holds, we can use the

embedding lemma (Lemma 4), the dual estimates (Lemma 8

and Lemma 9), and Theorem 1 to obtain the boundedness of

ei0(|D|) from MSP * to MJ'"2. □

Pi 'Hi

'Pl'll'

[7] A. Benyi, K. Grochenig, C. Heil, and K. Okoudjou, "Modulation spaces and a class of bounded multilinear pseudodifferential operators," Journal of Operator Theory, vol. 54, no. 2, pp. 387399, 2005.

[8] J. Chen, D. Fan, and L. Sun, "Asymptotic estimates for unimod-ular Fourier multipliers on modulation spaces," Discrete and Continuous Dynamical Systems A, vol. 32, no. 2, pp. 467-485, 2012.

[9] Q. Deng, Y. Ding, and L. Sun, "Estimate for generalized uni-modular multipliers on modulation spaces," Nonlinear Analysis: Theory, Methods and Applications, vol. 85, pp. 78-92, 2013.

[10] W. Guo and J. Chen, "Strichartz estimates on a-modulation spaces," Electronic Journal of Differential Equations, no. 118, pp. 1-13, 2013.

[11] H. G. Feichtinger and G. Narimani, "Fourier multipliers of classical modulation spaces," Applied and Computational Harmonic Analysis. Time-Frequency and Time-Scale Analysis, Wavelets, Numerical Algorithms, and Applications, vol. 21, no. 3, pp. 349359, 2006.

[12] H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhauser, Boston, Mass, USA, 1983.

[13] J. S. Han and B. X. Wang, "a-modulation spaces (I) scaling, embedding and algebraic properties," http://www.arxiv.org/ abs/1108.0460.

[14] G. P. Zhao, J. C. Chen, D. S. Fan, and C. Guo, "Unimodular Fourier multipliers and complex interpolation spaces on amodulation spaces," submitted.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referee for helpful comments. This work is partially supported by the NSF of China (Grant no. 11271330).

References

[1] H. G. Feichtinger, "Modulation spaces on locally compact Abelian group," Tech. Rep., University of Vienna, 1983.

[2] P. Grobner, Banachraume Glatter Funktionen and Zerlegungsmethoden [Ph.D. thesis], University of Vienna, 1992.

[3] L. Li, "Some remarks on a multiplier theorem of Hormander," Mathematica Applicata, vol. 15, pp. 152-154, 2002.

[4] T. Mizuhara, "On Fourier multipliers of homogeneous Besov spaces," Mathematische Nachrichten, vol. 133, pp. 155-161,1987.

[5] A. Benyi, K. A. Gröchenig, and L. G. Rogers, "Unimodular Fourier multipliers for modulation spaces," Journal of Functional Analysis, vol. 246, no. 2, pp. 366-384, 2007.

[6] A. Miyachi, F. Nicola, S. Rivetti, and N. Tomita, "Estimates for unimodular Fourier multipliers on modulation spaces," Proceedings of the American Mathematical Society, vol. 137, no. 11, pp. 3869-3883, 2009.

Copyright of Journal of Function Spaces is the property of Hindawi Publishing Corporation and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use.