Inclusion Relations between -Modulation Spaces and Triebel–Lizorkin Spaces

Abstract

In this paper, we obtain conditions of the inclusion relations between -modulation spaces and Triebel–Lizorkin spaces.

1. Introduction

The modulation space was first introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. modulation space has aclose relationship with the topics of time-frequency analysis (see [2]), and it has been regarded as a appropriate space for the study of partial differential equations (see [3–5]).

The -modulation space is introduced by Gröbner [6] to link Besov and modulation spaces by the parameter . One can find some basic properties about -modulation spaces in [7, 8]. Among many features of the -modulation spaces, an interesting subject is the inclusion between -modulation and function spaces, have been concerned by many authors to this topic, see [8–11]. As applications, -modulation spaces are quite recently applied to the field of partial differential equations. In [12], Misiolek and Yoneda proved locally ill-posedness of the Euler equations in the frame of -modulation spaces. Furthermore, Han and Wang [13] proved a global well-posedness for the nonlinear Schrödinger equations on -modulation spaces, and also in [14] studied the Cauchy problem for the derivative nonlinear Schrödinger equation on -modulation spaces.

Remark 1. Modulation spaces are special -modulation spaces in the case , so our theorems also works well in the special case .

In this research, we are interested in studying the inclusion relations between -modulation spaces and Triebel–Lizorkin spaces for , which greatly improve and extend the results for the inclusion relations between local Hardy spaces and -modulation spaces obtained by Kato in [10].

2. Preliminaries

The notation denotes the statement that with a positive constant that may depend on . The notation means the statement , and the notation stands for . For a multi-index , we denote , and .

Let be the Schwartz space and be the space of tempered distributions. We define the Fourier transform and the inverse Fourier transform of by

We give some definitions and properties of sequences.

Definition 2. Let . Let be a sequence, we denote its (quasi-) norm

and let be the (quasi-) Banach space of sequences whose (quasi-) norm is finite.

Let be a sequence, we denote its (quasi-) norm

and let be the (quasi-) Banach space of sequences whose (quasi-) norm is finite.

Let be a sequence, we denote its (quasi-) norm

and let be the (quasi-) Banach space of sequences whose (quasi-) norm is finite.

We recall some embedding lemmas about sequences defined above.

Lemma 3 (sharpness of embedding, for uniform decomposition). Suppose . Then

holds if and only if

Lemma 4 (sharpness of embedding, for dyadic decomposition). Suppose Then

holds if and only if

Lemma 5 (sharpness of embedding, for -decomposition). Suppose , . Then

holds if and only if

We recall some definitions of the function spaces treated in this paper.

Suppose that and are two appropriate constants, which relate to the space dimension , and a Schwartz functions sequence satisfies

Then constitutes a smooth decomposition of . The frequency decomposition operators associated with theabove function sequence are defined by

for . Let Then the -modulation space associated with the above decomposition is defined by

with the usual modifications when . For simplicity, we denote , and .

Remark 6. We recall that the above definition is independent of the choice of exact (see [8], proposition 2.3). Also, for sufficiently small , one can construct a function sequence such that =1 and if , when lies in the ball (see [15, 9, Appendix A]).

To define the Bosov spaces and Triebel–Lizorkin spaces, we introduce the dyadicdecomposition of . Let be a smooth bump function supported in the ball and be equal to 1 on the ball . For integers , we define the Littewood–Paley operators

Let and . Then the Besov spaces is defined by

Let , and . Then the Triebel–Lizorkin spaces is defined by

Let be the collection of all cubes in with sides parallel to the axes, centered at , and with side length , where and .

Let be a cube in and , then is the cube in concentric with and with side length times the side length of . We write if and

Let , then and stands for the largest integer less than or equal to .

Definition 7 (see [16]). Let , . Let and be integers with

 (1) The (complex-valued) function is called a -atom if  for some and  (2) Let . The (complex-valued) function is called a -atom if (20) is satisfied,  and  (3) The distribution is called an -atom if  for some and , where is a -atom and are complex numbers with  with usual modification if .

Lemma 8 (see [16]). Let , . Let and be fixed integers satisfying (19). Then is an element of if and only if it can be represented as
where are -atoms, are -atoms, and are complex numbers with

We also give the following lemma for inclusion relations between Besov and -modulation spaces [8].

Lemma 9. Let , and . Then the following tow statement are true:
(1) .(2) .

Lemma 10 (Young’s inequality). (1) Let . We have for all and , where independent of .(2)Let satisfy . Then we have

The following Bernstein multiplier theorem will be used in our proof.

Lemma 11 (Bernstein multiplier theorem). Let , for. Then,

3. Main Results

Now, we state our main results as follows.

Theorem 12. Let , , , and . Then, holds if and only if either of the following conditions is satisfied.

(1) ;(2) .

Theorem 13. Let , , , and . Then, holds if and only if either of the following conditions is satisfied.

(1) ;(2) .

We prove the following two propositions used for the proof of the Theorem 12.

Proposition 14. Let , , , and . Then we have

(1) (2).

Proof. Take to be a smooth function whose Fourier transform has compact small support, denote . We define

For a truncated (only finite nonzero items) nonnegative sequence , where N is some large integer.

By the definition of -modulation space , we have

On the other hand, we use the orthogonality of as , we obtain

Hence,

Thus, we obtain , if hold.

On the other hand, we obtain if hold.

Proposition 15. Let , , and . Then we have

Proof. Take to be a nonzero smooth function whose Fourier transform has small support, such that and if , where we denote . Denote

For a truncated (only finite nonzero items) nonnegative sequence . We have

On the other hand,

Hence,

Thus, we obtain , if .

Proof of Theorem 12. We divide this proof into two parts.

Necessary. For , using Proposition 14 and Lemma 3 to deduce .

On the other hand, we use Proposition 15 and Lemma 5 to deduce , which implies .

For , using Proposition 14 and Lemma 3 to deduce , which implies .

Sufficiency. For . We have , and then . using Lemma 9, we obtain . Thus we deduce that

Which is the desired conclusion.

For . Use obtained above to deduce that

for any .

We prove the following two propositions used for the proof of the Theorem 13.

Proposition 16. Let , , , and . Then we have

(1) (2)

Proof. Let be a nonzero Schwartz function whose Fourier transform has compact support in , satisfying on . Set . By the definition of , we have for , and if . Denote

we have for , where is a sufficiently large number. We define

for a truncated (only finite nonzero items) nonnegative sequence .

We first prove that the inclusion implies . By the definition of -modulation space, we obtain that

On the other hand, we turn to the estimate of , using the orthogonality of as , we obtain

Hence,

Thus, if , we obtain the desired inclusion

Next we prove that the inclusion implies . By the definition of -modulation space, we obtain that

Hence,

On the other hand, by the same argument of the previous proof, we deduce that

Thus, if holds, we obtain the desired inclusion

Proposition 17. Let . We have the following inclusion relation:

Proof. We first verify

for any -atom . Tack to be an -atom as in Definition 7 (with ). Observing that , we have

for . By the Bernstein multiplier theorem, we have the following estimate of :

Next, we turn to the estimate of an -atom for . By Definition 7, an -atom can be represented by

for some and , where are -atoms and are complex numbers with

for a fixed , we denote

Then, can be represented by

We now concentrate on the estimate of . By Definition 7, we have.

for . Recalling , we use (60) and the almost orthogonality of to deduce that

for all . By the Bernstein multiplier theorem, we deduce that

By a dilation argument, we have

Thus,

By Lemma 8, we have

which is the desired conclusion.

Proof of Theorem 13. We divide this proof into two parts.

Sufficiency. For , by Lemma 9, we obtain . Using , we deduce that

In addition, we have by Proposition 17. By potential lifting, we obtain

Thus, the desired conclusion can be deduced by a standard interpolation argument between (66) and (67).

For , recalling obtained in Proposition 17, we deduce that

for any .

Necessity. We use Proposition 16 to deduce inclusion relation . Then, Lemma 4 yields that for , while the inequality is strict for .

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by Tishreen University.