Journal of Function Spaces
Volume 2016 (2016), Article ID 1710260, 6 pages
http://dx.doi.org/10.1155/2016/1710260

Research Article

Trace Operators on Wiener Amalgam Spaces

Jayson Cunanan and Yohei Tsutsui

Department of Mathematical Sciences, Faculty of Science, Shinshu University, Asahi 3-1-1, Matsumoto, Nagano 390-8621, Japan

Received 24 February 2016; Accepted 27 March 2016

Academic Editor: HuoXiong Wu

Copyright © 2016 Jayson Cunanan and Yohei Tsutsui. 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.

Abstract.

The paper deals with trace operators of Wiener amalgam spaces using frequency uniform decomposition operators and maximal inequalities, obtaining sharp results. Additionally, we provide the embedding between standard and anisotropic Wiener amalgam spaces.

1. Introduction

The aim of this paper is to study the trace problem: what can be said about the trace operator ,

as a mapping from to We note that, for a tempered distribution defined on has no straightforward meaning and the question is how to define the trace for a class of tempered distributions. One can resort to the Schwartz function , which has a pointwise trace . It can be extended to (quasi-)Banach function spaces which contain the Schwartz space as a dense subspace.

Our setting is on Wiener amalgam spaces. These spaces, together with modulation spaces, were introduced by Feichtinger [1–3] in the 80s and are now widely used function spaces for various problems in PDE and harmonic analysis [4–10]. They resemble Triebel-Lizorkin spaces in the sense that we are taking norms but differ with the decomposition operator being used. Instead of the dyadic decomposition operators used for Triebel-Lizorkin spaces, Wiener amalgam spaces use frequency uniform decomposition operators , where denotes a unit cube with center and

The concept of trace operator plays an important role in studying the existence and uniqueness of solutions to boundary value problems, that is, to partial differential equations with prescribed boundary conditions [11, 12]. The trace operator makes it possible to extend the notion of restriction of a function to the boundary of its domain to “generalized” functions in various function spaces with regularity. Now, we give a formal definition for the trace operators.

Definition 1. Let and be quasi-Banach function spaces defined on and , respectively. Assume that the Schwartz class is dense in Denote

Assuming that there exists a constant such that

one can extend by the density of in and we write , which is said to be the trace of . Moreover, if there exists a continuous linear operator such that is the identity operator on , then is said to be a trace-retraction from onto

For (-) modulation spaces, Besov spaces, and Triebel-Lizorkin spaces, trace theorems have been extensively studied [12–14]. Feichtinger et al. [13] considered the trace theorems on anisotropic modulation spaces with and they obtained . In [15, 16], we find that, for , and , we have and (the case is omitted). The use of atoms as a framework in studying trace problems can be found in [16] and the references within.

Our main results are the following.

Theorem 2. Let Then

is a trace-retraction from to

In view of the embedding in Theorem 6(II-ii), we immediately have the following corollary.

Corollary 3. Let Then for any

We remark that Corollary 3 is an improvement of an older trace theorem found in [14] and that our result is sharp at least for Moreover, our result shows independence of . This is due to the pointwise estimates we were able to prove in Section 3. An interesting observation is that the trace theorem of Triebel-Lizorkin spaces stated above shows independence in . This difference might be due to the decomposition operators used in the norm of each of the function spaces.

The paper is organised as follows. In Section 2, the embedding between standard and anisotropic Wiener amalgam spaces is given. We also define notations, function spaces, and some lemmas to be used throughout this paper. In Section 3, we prove our main result, Theorem 2, and the sharpness of Corollary 3.

2. Preliminaries

Notations. The Schwartz class of test functions on will be denoted by and its dual and the space of tempered distributions will be denoted by . norm is given by whenever and . The Fourier transform of a function is given by

which is an isomorphism of the Schwartz space onto itself that extends to the tempered distributions by duality. The inverse Fourier transform is given by . Given , we denote by the conjugate exponent of (i.e., ). We use the notation to denote for a positive constant independent of and . We write and . We now define the function spaces in this paper.

Let be a smooth bump function satisfying

We write, for and ,

Put

Definition 4 (Wiener amalgam spaces). For , and , the Wiener amalgam space consists of all tempered distributions for which the following is finite:

with .

We note that (10) is a quasi-norm if and norm if . Moreover, (10) is independent of the choice of . We refer the reader to [1, 2, 17] for equivalent definitions (continuous versions).

We write and define the anisotropic Wiener amalgam spaces by the following norm:

Similarly, for , we define

Comparing amalgam spaces with anisotropic amalgam spaces we see that is rotational invariant but is not. Using the almost orthogonality of we see that is independent of . Moreover, recalling that is the function sequence equipped with the norm, it is easy to see that is a quasi-Banach space for any and a Banach space for any Moreover, the Schwartz space is dense in if The proofs are similar to those of amalgam spaces in [1, 2, 17].

We collect properties of Wiener amalgam spaces in the following lemma.

Lemma 5. Let for and for Then one has the following:

(1)(2) is dense in if and (3)if and , then (4)if , then (5)(complex interpolation) let , , , and Then

The proofs of these statements can be found in [1, 3, 17, 18].

Theorem 6 (embedding: ). Let and .

(I)The case .

(I-i)The case

(I-ii)The case

(II)The case .

(II-i)The case . If , then

 for any .(II-ii)The case . If , then

 for any .

(III)The case .

(III-i)The case

(III-ii)The case

Proof. For part (I), it suffices to show the following estimates.

(I-i) Consider

(I-ii) Consider

(II-i)Let . We may assume that :

 The last term is equivalent to

 where and have been used.(II-ii)Let . It suffices to show the embedding in the case . Remark that and . Let :

 Here, we have used . Because , and ,

(III-i) Consider

 Here, we have used .(III-ii)Using the embedding ,

 In the last inequality, we need .

Lemma 7 (Triebel [12]). Let and . Let be a sequence of compact subsets of Let be the diameter of If , then there exists a constant such that

holds for all , where , , and

Definition 8 (maximal functions). Let and . Then

Proposition 9. Let and . Then

are equivalent norms in and , respectively.

The proof is a direct consequence of Lemma 7, taking . See also [14, Proposition].

3. Proof of the Main Results

First, we narrate the idea of the proof. We give an equivalent formulation for , a function in , via some , a function in Then we compute for pointwise estimates between the corresponding norms and norms for cases and , separately. Finally, taking norms and using our equivalent norms in Proposition 9, we arrive to our conclusion.

We denote by the partial (inverse) Fourier transform on . Write as versions of (9) in . By the support property of , we observe

where , and . Note that the left-hand side is a function in while the right-hand side is a function in .

Recall our maximal function (30) and take ; we have, for ,

Proof of Theorem 2. We start by taking the -norm of (33). We write

For , we estimate (35) by

Note that , where is the th column of the identity matrix. In the sequel, it suffices to consider only the case Moreover, we write for some satisfying (9). Using (34) we have

Combining (36) and (37), then taking the -norm, and raising to th power give

Integrating over ,

Note that the last inequality follows from Proposition 9.
For , we use Minkowski’s inequality to give an upper bound of (35) as follows:

Repeating the arguments above on (40) gives us the estimate

Hence, we arrive to our desired estimates.
Let be a function with and For any , we define We easily see that and when . Moreover, we can decompose due to the way is defined in (9). Now we do an estimate

Thus, .

As the end of this paper, we discuss the optimality of Corollary 3. We recall the counterexample given in [13]. For , there exists a function which shows

Since , we also have . Hence, Corollary 3 is sharp for (refer to Figure 1). We now claim that it is also sharp for all . Contrary to our claim, suppose implies . Then, by interpolation with the estimate for a point with , one would obtain an improvement for the segment connecting and (refer to Figure 2), which is not possible.

(a)

(b)

Figure 1: Comparison between the critical regularity index for (a) and (b).

Figure 2: Contradiction argument using interpolation.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by JSPS, through “Program to Disseminate Tenure Tracking System.” Yohei Tsutsui was also partially supported by JSPS, through Grant-in-Aid for Young Scientists (B) (no. 15K20919).

References

H. G. Feichtinger, “Banach convolution algebras of wiener's type,” in Proceedings of the Conference Function, Series, Operators, Colloquia Mathematica Societatis János Bolyai, pp. 509–524, North-Holland, 1980.

H. G. Feichtinger, “Banach spaces of distributions of Wiener's type and interpolation,” in Functional Analysis and Approximation, vol. 60, pp. 153–165, Birkhäuser, Basel, Switzerland, 1981.

H. G. Feichtinger, “Modulation spaces on locally compact abelian groups,” Tech. Rep., University of Vienna, Vienna, Austria, 1983.

R. Bělohlávek, Fuzzy Relational Systems: Foundations and Principles, Kluwer Academic/Plenum Press, New York, NY, USA, 2002.

A. Benyi and T. Oh, “Modulation spaces, Wiener amalgam spaces, and Brownian motions,” Advances in Mathematics, vol. 228, no. 5, pp. 2943–2981, 2011.

E. Cordero and F. Nicola, “Strichartz estimates in Wiener amalgam spaces for the Schrödinger equation,” Mathematische Nachrichten, vol. 281, no. 1, pp. 25–41, 2008.

E. Cordero and F. Nicola, “Remarks on Fourier multipliers and applications to the wave equation,” Journal of Mathematical Analysis and Applications, vol. 353, no. 2, pp. 583–591, 2009.

J. Cunanan, M. Sugimoto, and M. Kobayashi, “Inclusion relations between -Sobolev and Wiener amalgam spaces,” Journal of Functional Analysis, vol. 419, no. 2, pp. 738–747, 2014.

K. Grochenig and C. Heil, “Modulation spaces and pseudodifferential operators,” Integral Equations and Operator Theory, vol. 34, no. 4, pp. 439–457, 1999.

B. Wang and C. Huang, “Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations,” Journal of Differential Equations, vol. 239, no. 1, pp. 213–250, 2007.

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, USA, 1998.

H. Triebel, Theory of Function Spaces, Birkhäauser, Boston, Mass, USA, 1983.

H. G. Feichtinger, C. Huang, and B. Wang, “Trace operators for modulation, α-modulation and Besov spaces,” Applied and Computational Harmonic Analysis, vol. 30, no. 1, pp. 110–127, 2011.

H. Triebel, “Modulation spaces on the Euclidean n-space,” Zeitschrift für Analysis und ihre Anwendungen, vol. 2, no. 5, pp. 443–457, 1983.

M. Frazier and B. Jawerth, “Decomposition of Besov spaces,” Indiana University Mathematics Journal, vol. 34, no. 4, pp. 777–799, 1985.

H. Triebel, Theory of Function Spaces II, Birkhäuser, Basel, Switzerland, 1992.

H. G. Feichtinger, “Generalized amalgams, with applications to Fourier transform,” Canadian Journal of Mathematics, vol. 42, no. 3, pp. 395–409, 1990.

J. Toft, “Continuity properties for modulation spaces, with applications to pseudo-differential calculus-II,” Annals of Global Analysis and Geometry, vol. 26, no. 1, pp. 73–106, 2004.