Inclusion Relations between -Modulation Spaces and Triebel–Lizorkin Spaces

e modulation space 푠 푝,푞 was first introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. modulation space has a close 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]). e -modulation space is introduced by Gröbner [6] to link Besov and modulation spaces by the parameter 0 ≤ 훼 ≤ 1. 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.


Introduction
e modulation space 푠 푝,푞 was first introduced by Feichtinger [1] in 1983 by the short-time Fourier transform. modulation space has a close 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][4][5]). e -modulation space is introduced by Gröbner [6] to link Besov and modulation spaces by the parameter 0 ≤ 훼 ≤ 1. 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][9][10][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 훼 = 0, so our theorems also works well in the special case 훼 = 0.
In this research, we are interested in studying the inclusion relations between -modulation spaces 푠,훼 푝,푞 and Triebel-Lizorkin spaces 푝,푟 for 푝 ≤ 1, which greatly improve and extend the results for the inclusion relations between local Hardy spaces and -modulation spaces obtained by Kato in [10].
Suppose that 푐 > 0 and 퐶 > 0 are two appropriate constants, which relate to the space dimension , and a Schwartz functions sequence 훼 푘 푘∈ℤ satisfies en 훼 푘 푘∈ℤ constitutes a smooth decomposition of ℝ . e frequency decomposition operators associated with the above function sequence are defined by for 푘 ∈ ℤ . Let 0 < 푝, 푞 ≤ ∞, 푠 ∈ R, and 훼 ∈ [0, 1). en the -modulation space associated with the above decomposition is defined by Remark 6. We recall that the above definition is independent of the choice of exact (see [8], proposition 2.3). Also, for sufficiently small 훿 > 0, one can construct a function [15, 9, Appendix A]).
Let be a cube in ℝ and 푚 > 0, then is the cube in ℝ concentric with and with side length times the side Let 푎 ∈ ℝ, then 푎 + = max(푎, 0) and [푎] stands for the largest integer less than or equal to .

Main Results
Now, we state our main results as follows.
We prove the following two propositions used for the proof of the eorem 12.

Journal of Function Spaces 4
Proof. Take to be a nonzero smooth function whose Fourier transform has small support, such that ✷ 푓 = 푓 and ✷ ℓ 푓 = 0 if 푘 ̸ = ℓ, where we denote 푓 Which is the desired conclusion.
By the definition of -modulation space 푠,훼 푝,푞 , we have On the other hand, we use the orthogonality of ∈ℤ as 푁 → ∞, we obtain (32) Proof. Let be a nonzero Schwartz function whose Fourier transform has compact support in {휉  . We first prove that the inclusion . By the definition of -modulation space, we obtain that On the other hand, we turn to the estimate of ᐉ ᐉ ᐉ ᐉ ᐉ ᐉ ᐉ ᐉ 푝,푟 , using the orthogonality of as 푁 → ∞, we obtain (42)