Cheng and Tao Journal of Inequalities and Applications (2015) 2015:344 DOI 10.1186/s13660-015-0872-4

O Journal of Inequalities and Applications

a SpringerOpen Journal

Fractional integral operator on modulation and Wiener amalgam spaces

CrossMark

Meifang Cheng* and Wenyu Tao

Correspondence:

cmf78529@mail.ahnu.edu.cn Department of Mathematics and Computer Science, Anhui Normal University, Wuhu, 241000, P.R. China

£ Springer

Abstract

The purpose of this paper is to investigate the mapping properties of the fractional integral operators on weighted modulation spaces. Based on this result, we also study the boundedness of the bilinear fractional integral operators on product Wiener amalgam spaces. Our results show that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces.

MSC: Primary 42B20;42B25

Keywords: fractional integral operator; weighted modulation space; bilinear fractional integral operator; Wiener amalgam space

1 Introduction

Time-frequency analysis is a modern branch of harmonic analysis. It has many applications in signal analysis and wireless communication (see [1, 2]). Time-frequency analysis is a form of local Fourier analysis that treats time and frequency simultaneously. Inspired by this idea, modulation and Wiener amalgam spaces have been introduced and used to measure the time-frequency concentration of a function or a tempered distribution (see [3-7]). During the last ten years, these two function spaces have not only become useful function spaces for time-frequency analysis, they have also been employed to study boundedness properties of pseudo-differential operators, Fourier multipliers, Fourier integral operators, and well-posedness of solutions to PDEs. For more details of the applications of these two function spaces, the reader is referred to [8-19] and the references therein.

In this paper, we are mainly concerned with the mapping properties of the fractional integral operator on weighted modulation spaces. Using this result, we also prove the bound-edness of the bilinear fractional integral operator on product Wiener amalgam spaces. From our results, we will see that, besides modulation spaces, Wiener amalgam spaces are good substitutions for Lebesgue spaces.

The fractional integral operator Ia is defined by

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where 0 < a < n. The well-known Hardy-Littlewood-Sobolev theorem says that Ia is bounded from ¿p(Mn) to ¿q(Mn) for 1 < p < q < œ and 1 = p - n. This theorem plays important roles in partial differential equations.

In recent years, many authors were interested in the mapping properties of the fractional integral operator on modulation spaces. For example, in [20], Tomita regarded the operator Ia as a special case of pseudo-differential operator and proved the following results. Set 0< a < n, 0 < e <1- ", and 0 < p < " .If 1 < p1, p2, q1, q2 < œ satisfy

11 a 11 a

q1 P1 " 5 q2 P2 " P'

then there exists a constant Ca>s>p > 0, such that

lllf \Mq1,q2 (Rn) < Ca,s,p |f ||Mp1,p2(Rn) for allf e S(Mn).

Subsequently, Sugimoto and Tomita improved the results in [20] and obtained necessary and sufficient conditions for the boundedness of the operator Ia (see [21]). Using the definitions of the discrete form for modulation spaces, which will be given by Definition 2.4 in the next section, they proved that Ia is bounded from Mp1,q1(Mn) to Mp2,q2(Mn) if and only if

11 a 11 a

p2~ p1 n ' q2 q1 n '

where 0 < a < n and 1 <pi,p2, qi, q2 < œ.

Recently, in [22], by using the norm of Hardy spaces Hp, Chen and Zhong introduced the modulation Hardy spaces Mp,q for 0 < p < 1 and 0 < q <œ. They proved that if n—< r <œ, then Ia is bounded from M-a to Mr,q for 0 < p < 1 and 0 < q <œ. Moreover, for 1 <p < œ and 0 < q1, q2 < œ, they obtained Ia is bounded from Mp,q1 to Mœ,q2 if and only if

I a 1 1 a p n q2 q1 n

In their proofs, Chen and Zhong also used the definition of the discrete form for modulation spaces.

Inspired by Sugimoto and Tomita, using the definition of integral form for modulation spaces, which will be given by Definition 2.3 in Section 2, we prove the following result.

Theorem 1.1 For 0 < a < n and s1, s2 e R, let 1 <p1,p2 < œ and 0 < q1, q2 <œ. If

II a 11 a ( 1 1 \

— <---, —< — + — and s2 < s1 + n---,

p2 p1 n q2 q1 n q1 q2

then the fractional integral operator Ia is bounded from MS11,q1(Mn) to Mp22,q2(Mn).

Another purpose of this paper is to study the mapping properties of the bilinear fractional integral operator Ba, which is defined by

„ ,MM f f(x -y)g(x + y) ,

B f' |y|-

on modulation spaces. It was showed in [22] that if 1 <p < ^na and 0 < q <to, then < C\f |imi,i\\F-\)\\MP.q.

In particular, for p < q,

iB f, g )| MTO,to < C\f\\mi,i \\g\\Mqp.

Inspired by Chen and Zhong, in this paper, we investigate the boundedness of the bilinear fractional integral operator Ba on Wiener amalgam spaces. Our result is as follows.

Theorem 1.2 For 0 < a < n, 1 <p0,p1 < to, and 1 < q0, q1,p2, q2 < to, letp'0 and q'0 denote the conjugate index ofp0 and q0, respectively. Suppose q'0 > p'0, p^ < p^ - ^^, < + ^, and qr + p^ = + then the operator Ba maps W(FLp1,Lq1) x W(FLp2,Lp0) to W (FLq2, Lq0).

In what follows, we always denote C to be a positive constant that may be different at each place, but is independent of the essential variables.

This paper is organized as follows. In Section 2, we give the definitions and basic properties of modulation and Wiener amalgam spaces. Section 3 is devoted to the proofs of our main results.

2 Basic definitions and important lemmas

The following notations will be used throughout this paper. Let S(Rn) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on Rn and S'(Rn) be the topological dual of S(Rn). For a function f in S(Rn), its Fourier transform is defined by f (a) = /f (t)e-2niat dt, and its inverse Fourier transform is f (t) =/(-t). The translation and the modulation operators are defined by

Txf (t)=f (t - x) and Maf (t) = e2n iatf (t)

for every x, a e Rn. For s e R and x e Rn, the weight function (x)s = (1 + |x|2)2.

Definition 2.1 Let g be a non-zero Schwartz function and 1 < p, q <to and s e R, the weighted modulation space Mp,q(Rn) is defined as the closure of the Schwartz class with respect to the norm

If I Imp» =(f (f \Vgf (x, w)\pd^\" <w>sqdwV,

s \Jw\ J Rn / /

with obvious modifications for p or q = œ, where Vgf (x, w) is the so-called short time Fourier transform (STFT), which is defined by

Vgf (x, w) = <f, Mf Txg) = j e-2niwyf (y)g(FX) dy,

i.e. the Fourier transform F applied to fTxg.

Recently, the above definition has been generalized by Kobayashi in [23] to the case 0 < p, q < œ. In his definition, the function g is restricted to the space $S (R"), which is defined as follows.

Definition 2.2 For S > 0, we define ®S (R") to be the space ofallg e 5(R") satisfying suppg c {f : if |< 1} and J2 - Sk) = 1.

We may choose a sufficiently small S, such that the function space $S (R") is not empty.

Definition 2.3 Given a g e $S(R"), and 0 < p, q <œ, s e R, we define the modulation space Mp,q(R") to be the space of all tempered distributionsf e S'(R") such that the quasinorm

If WrnP-q =(f (f f * (Mwg)(x)|pd^p <w)sqd^q \Jr"\ J R" / /

is finite, with obvious modifications forp or q = œ.

For a more general definition, involving different kinds of weight functions, both in the time and the frequency variables we refer the reader to [24]. Definitions 2.1 and 2.3 are the integral form. We also have the definition of discrete form for modulation spaces, which is very useful in studying unimodular Fourier multipliers.

Definition 2.4 Let 1 < p, q <œ, s e R, and $ e S (R") be such that supp $ c [-1,1]", - k) = 1

for all f e R". Denote $k(f ) = $(f - k) and let $k(D) be the Fourier multiplier operator given by $k(D)f (f ) = $k(f )f(f ). Then the weighted modulation space Mp,q(R") consists of all f e S'(R") such that

If IlMp,q = (E (1 + lkl)sq||$k(D)f lip)q < œ

with obvious modifications forp or q = œ.

There is yet another definition of modulation spaces, which is given by Gabor frames and plays a key role in studying simultaneously the local time and global frequency behavior of functions (see [24]).

The following basic properties of weighted modulation spaces, which play important roles in this article, can be found in [23, 25].

Lemma 2.5 Let 0 < p, q <to andg e (Rn). Then

(1) Different test functions gi, g2 e (Rn) define the same spaces and equivalent quasi-norms on Mp,q (Rn).

(2) Let 0 <p0 <pi <to, 0 < q0 < qi <to and s e R, then

Mp0,q0 (Rn) Mpi,qi (Rn).

(3) If 0 <p, q < to, then S(Rn) is dense in Mpq(Rn).

(4) For 1 < p, q < to and s e R, (Mfq)' = M-'/.

To prove our main results, we also need the definition of the Wiener amalgam space W (FLp, Lq)(Rn).

For 1 < p < to, let FLp(Rn) be the space of tempered distributions with their Fourier transforms in Lp(Rn), that is,

FLp(Rn) = f e S'(Rn) |f e Lp(Rn)}

with norm |f ||FLp = l|/llip(Rn).

Definition 2.6 For 1 < p, q < to, a tempered distribution f is in the Wiener amalgam spaces W(FLp,Lq)(Rn), iff is locally in FLp(Rn), that is, for every non-zero g e Cg°(Rn), FfTxg) e Lp (Rn), and

llf llw(Fip,Lq) = ( f (i \F(fTxg)(y)\pdy\ d^

\Jw\ J Rn / /

is finite, with obvious modifications for p or q = to. This definition is independent of the choice of the test function g e CTO(Rn).

Both the modulation spaces and the Wiener amalgam spaces are mixed-norm function spaces. The following lemma gives the relationship between them.

Lemma 2.7 Let F be the Fourier transform and 1 < p, q <to, then Mp,q(Rn) = FW(FLp, Lq)(Rn).

Proof Chooseg e S(Rn) such thatg = 0 andf e S'(Rn). For every x, % e Rn, the definition of the short time Fourier transform implies that

Vgf (x, %) = e-2n x% Vf (% ,-x)

V/ (%, -x) = F (f T% g)(-x).

Therefore, Definitions 2.1 and 2.6 yield

If\\Mpq = (f (f \Vgf(x,£)\pdÀPdï

\jRn\ JR" /

7 (7 \Vgm,-x)\pd^"df)

\JR" \JR" / /

7 (f \F tfT£g)(-x)\pdxY d£

\JRn\ J R" /

= \\J\\W(FLP,L<1)-

The proof of Lemma 2.7 is completed. □

3 Proof of the main results

In this section, we are going to prove our main results. First, we show the proof of Theorem 1.1.

1 1 n 2 r( — )2a

Set Ka (x) = Y^jy ■jxj1^^, where y (a) = r(n-") . The fractional integral operator Ia, which

is defined by Ia(f)(x) = (Ka * f)(x), may be realized on the transform side as a Fourier

multiplier

Cf)(£ ) = ma (f)f (S), where ma (S )= K (S).

Proof We consider the following three cases to obtain Theorem 1.1.

Case 1: — = — - - and q1 > q2.

P2 P1 " 11 12

S<5/in> n\

Let g, x e ^ (Rn) satisfy the condition g = x * x. Then M£ g = M^x * M^x. Young's inequalities and the Hardy-Littlewood-Sobolev theorem give

ia ( ) , ,P2,q2 =

i \\laf) *M£g||®2 <£>q2S2 d£

f | \K *f * M£x i*|Mçx I \\qp2 <£>q2S2 d£

< CIIx II7/ Ilf *Mçx I\qp1 <£>q2S2 dAq2.

\J R" /

Using the Holder inequalities for the exponent ^ and (^y, we get

32 J_ 31 \ q1 q2

IIiaf ImP232 < CIx IIJ f IIf * Mçx II3p1 <£>32S1 <£>32(s2-s1) d£

S2 \J R"

< CIx IIJ f (If * Mçx II m <£ >s1 )32 31 d£

( i 31-32

f S s, ) 3132 \ 32 31 <£>(s2-s1)31-32 d^

Denote

J =(f <%>(s2-si)q-22dfq2 qi

Since s2 < si + n(-i - -i), an easy computation shows that

i J )<%>(s2-si)™ d%

Ji%i<i Ji%i>i.

2i-22 2122

< Cn +(f i°°P(s2-si) ^ Pn-i dp d%

\Jsn-1 Ji

< Cn < to. Therefore,

ll/f hm^« < C^|f iMfiqi. Case 2: — = — - - and qi < q2. Since 0 < a < l, we can choose 0 < q2 <to such that

p2 pi n i 2 n 2

qi > q2 and i < -1 + a .According to the proof of case l, we can see that/« is bounded from

q2 ql n

Mpi,qi to mp22,22 . By the condition q2 > qi > q2 and the fact Mf22,q~2 Mf22,q2, the proof of Case 2 is completed.

Case 3: — < — - a .In this case, 0 < — - a < l. Take l < p2 < to such that i = — - a,

p2 pi n pi n 2 p2 pi n

then p2 > p2. Using the embedding result Mp2,q2 Mp22,q2 and the proof of Case l and 2, we finish the proof of Theorem l.l. □

Now, we turn our attention to the proof of Theorem l.2. From a Fourier point of view, the bilinear fractional integral operator can be rewritten as

{BOfg))(% ) = / g(t)F (t, % )e-2n it% dt,

F (t, %)=/ yrnf(t - 2y)e2n iy% dy.

For the proof of Theorem l.2, we need some lemmas.

Lemma 3.1 Let H(t, y) = e-2nipyF^1f (y) and H2 be the Fourier transform for the second variable of H, thenH2(t, %) =f (t + %) andF(t,2%) = 2-a/n-a(H(t, • ))(%).

Proof We only prove the second equality. The first one is very easy, we omit the details here. Note that if we set y' = 2y, F(t, 2%) can be rewritten as

F (t, 2% ) =

f(t -/) e2n iy % 2-ndy'

= 2-a i lyla-nf(t + y)e-2niy% dy = 2-a(/n-aH(t, • ))(-%).

We finish the proof of Lemma 3.l. □

Lemma 3.2 Denote Ff(t, £) = F(t,2£), then \\Ff\\Mp q = Cn,p,q\\F\\mp'9, where Cn,p,q =

2-«(i+ 1).

Proof Takingg e 5(R2n) \ {0}, then we have

VgF2(x, £,y, n)= f F2(s, t)g(s - x, t - £)e-2ni(sy+tn) dsdt

= f F(s, 2t)g(s - x, t - £) e-2ni(sy+tn) dsdt.

Set t' = 2t, then

V^f(x, £, 7, n) = 2-

5 -x, t—^ e-2ni(s7+ 2n) dsM

7 f(s,

2-" /" F (s, i(sy+t,2 dsdt'

= 2-«ViF(x,2£,),

where <(s, =g(s, 2) e 5(R2n) \ {0} is another window function. Definition 2.1 gives

\\f2 WMP,q =(f (f \VgF2(x, £, y, n)\pdxd£) 'dyd^

\J R2A J R2n

\J r2A JR2

ViF(x,2£,n

dxd£ ) dydn \ .

Let £' = 2£, n' = 2. It is easy to check that

l|F22\\MPq = 2-"(1+i-q)(f (i \ViF(x,£,7,n)|pdXd£)P dydn \,/r2A ./r2" /

= 2-"(1+ p-q)||F ||Mm = C„,M||F .

We complete the proof of Lemma 3.2.

Lemma 3.3 Suppose <o, g, h to be the non-zero Schwartz functions and W (g, h)(t, s) = g(t)h(s)e2nits, then

Vw (<0,<1)W (g, h)(uo, u, vo, v) = e-2niuouV<0g (uo, u - vo)V< h(u, v - uo).

Proof The definition of the short time Fourier transform yields

Vw (<o,<1) W (g, h)(uo, u, vo, v)

= f f W(g, h)(t,s) W(<o, <i)(t - uo,s - u)e-2ni(tvo+sv) dtds jRn JRn

= f f g(t)h(s)e2nits<o(t- uo)<i(s - u)e2ni(t-«o)(s-«)e-2ni(tvo+sv) dtds

jRn JRn

_ lUQU

i g(t)<0

o(t - u°)e2n i(u-vo)tdt)l I h(s)<_(s - u)e

-2n i(v-u°)s

_ e-2niu°uV<°g(uo, u - vo)Vhh(u, v - uo).

The proof of Lemma 3.3 is completed.

Lemma 3.4 For 1 < po < to and 1 < qo,p2, q2 < to, let p'o and q'o denote the conjugate index ofpo and qo, respectively. Ifq'o > p'o and + p^ _ p^ + then W(g, h) defined in Lemma 3.3 is bounded from W (FLp2, Lp'°) x Mq2«° to Mp°,q°.

Proof For eachg e W(FLp2,Lp'o) and h e M«2^, Lemma 3.3 indicates

lW(g,h) Wmp'o^O

/ ( / I Vw (ho,h1) W (g, h)(uo, u, vo, v) |p° duo du) p° dvo dv

\J r2A J«2" /

I ( I |V<° g(uo, u - vo)|p° |V< h(u, v - uo)|p° duo duY° dvo dv

\J R2 A J R2n /

Note that V<°g(uo, u - vo) _ e-2niu°(u-v°)V<0g(u - vo,-uo). If we denote P(x,y) _ P(-x,y), then

lW (g, h)WMpU

/2 (/2 ^- vo,-uo)|po|V<1 h(u,v - u°)|p° duodu^dvodv

/ (/ |V<oii(vo- u,-u°)|p° |V<1 h(u, v - u°)|p° duo du) ° dvo dv

R2n R2n o

V (i (|V<50,-uo)|p'° *|V<1 h( •, v - u°)|p'°)(vo) du°) p'° dv

R2n Rn o

7 i (|V5°'i^^Uo)|p/° *|V<1 h( • ,v-u°)|p'°)(vo)du

Rn Rn o

p'o J \ «o

«o dv

Lp'° (v°)

Since 4- + 4- = — + —, using Minkowski's integral inequalities, Young's inequalities, and

q 0 p 0 p 2 q 2

Lemma 2.7, we obtain

lW (g, h)Uq°

I (i W | VjoiO^-uo) |p° * | V<1 h(• , v - u°) |p° W q° du°) p° dv

\JRn\ J Rn p° / ,

«o . J_

Lp° (v°)

The proof of Lemma 3.4 is completed.

We give the proof of Theorem 1.2. Proof Choose h e S(Rn) \ {0} and denote by <•, •) the inner product, then

(BJfTg), h) = / )(£)h£d£

= f f F(t, £)g(t)W)e-2nit£ dtd£

JR" JR"

= (F, W(g, h)),

where W(g, h) =g(t)h(£)e2n,t£. Holder's inequalities show \[BJfTg), h)\ = \(F, W(g,h))\

= \( VW(io,ii)F, VW(i0,ii)W(g,h)\ <WF\MPoqo |W(g,h)|Mp0,io.

Lemma 3.1, Lemma 3.2, and Theorem 1.1 yield

||FWmPom = Cn,po,qo ||F2 Wm^O = Cn,po,qo ^n-a (H(t, |Mp0,q0 < C«,po,qo |Hfe •)|MP1,i1 = C«,po,qo H^" = C«,po,qo |f W W(FLP1 ,Lq1).

^ |MP1,q1

On the other hand, Lemma 3.4 gives

II W(g, h)IU,qo < C|g ||w(FLP2,LP'o)||h||мq2,qo. Therefore,

\{Bafg), h)\ < C

n,po,qo |fW W(FLP1 ,Lq1)||g| w(fLp2 lpo) W^M®^ ,

which implies Ba (f,g) e Mq2,qo and

|Ba (f,g)|Mq2,qo < Cn,Po,qo |f Ww(FLP1,Lq1) |g| W(F^2 ,lp'o).

Lemma 2.7 indicates ||Ba(f,g)||W(FLq2 Lqo) = ||Ba(f,g)WMq2,q0,we conclude

|Ba (f, g)|W(FLq2,Lq0) < C«,Po,qo |f Ww (FLP1,Lq1)|g| W (FLP2L/0). The proof of Theorem 1.2 is finished. □

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The authors contributed equally to the writing of this paper. Allauthors read and approved the finalmanuscript

Acknowledgements

This work is supported by the National Nature Science Foundation of China (No. 11201003) and NNSF (No. KJ2014A087)

of Anhui Province in China.

Received: 8 July 2015 Accepted: 16 October 2015 Published online: 29 October 2015

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