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 . Thenare 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 observewhere , 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 writeFor , we estimate (35) byNote 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 haveCombining (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 estimateThus, .
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.
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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).