Trace operators on Wiener amalgam spaces

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 embeddings between standard and anisotropic Wiener amalgam spaces.


Introduction
The aim of this paper is to study the trace problem: What can be said about the trace operator T, T : f (x) → f (x, 0),x = (x 1 , x 2 , ..., x n−1 ), as a mapping from W p,q s (R n ) to W p,q s (R n−1 ). We note that for a tempered distribution f defined on R n , f (x, 0) 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 φ, which has a pointwise trace φ(x, 0). It can be extended to (quasi-)Banach function spaces which contain the Schwartz space S as a dense subspace.
Our setting is on Wiener amalgam spaces. These spaces, together with modulation spaces, were introduced by Feichtinger [5,6,7] in the 80's and are now widely used function spaces for various problems in PDE and harmonic analysis [1,2,3,11,16]. They resemble Triebel-Lizorkin spaces in the sense that we are taking L p (ℓ q ) norms, but differ with the decomposition operator being used. Instead of the dyadic decomposition operators ∆ k ∼ F −1 χ {ξ:|ξ|∼2 k } F used for Triebel-Lizorkin spaces, Wiener amalgam spaces use frequency uniform decomposition operators k ∼ F −1 χ Q k F, where Q k denotes a unit cube with center k and ∪ k∈Z n Q k = R n .
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 [4,14]. 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 regulariy. Now, we give a formal definition for the trace operators. Definition 1.1. Let X and Y be quasi-Banach function spaces defined on R n and R n−1 , respectively. Assume that the Schwartz class S is dense in X. Denote Assuming that there exist a constant C > 0 such that ||Tf || Y ≤ C||f || X , ∀f ∈ S, one can extend T : X → Y by the density of S in X and we write f (x, 0) = Tf, which is said to be the trace of f ∈ X. Moreover, if there exist a continuous linear operator T −1 : Y → X such that TT −1 is the identity operator on Y, then T is said to be a trace-retraction from X onto Y.
Our main results are the following. Then is a trace-retraction from W p,q,1∧q s (R n ) to W p,q s (R n−1 ).
In view of the embedding in Theorem 2.1 (II-ii), we immediately have the following corollary.
We remark that our result shows independence of p. This is due 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 q. This difference might be due to the decomposition operators used in the norm of each function spaces.
The paper is organised as follows: In Section 2, the embeddings between standard and anisotropic Wiener amalgam spaces are 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 1.1 and the sharpness of Corollary 1.1.

Preliminaries
Notations. The Schwartz class of test functions on R n shall be denoted by S := S(R n ) and its dual, the space of tempered distributions, by which is an isomorphism of the Schwartz space S(R n ) onto itself that extends to the tempered distributions S ′ (R n ) by duality. The inverse Fourier transform is given by we denote by p ′ the conjugate exponent of p (i.e. 1/p + 1/p ′ = 1). We use the notation u v to denote u ≤ cv for a positive constant c independent of u and v. We write a ∧ b := min(a, b) and a ∨ b := max(a, b). We now define the function spaces in this paper. Let η : R → [0, 1] be a smooth bump function satisfying We write for k = (k 1 , ..., k n ) and ξ = (ξ 1 , ..., ξ n ), Definition 2.1 (Wiener amalgam spaces). For 0 < p, q ≤ ∞, and s ∈ R, the Wiener amalgam space W p,q s consists of all tempered distributions f ∈ S ′ for which the following is finite: We note that (2) is a quasi-norm if 0 < p, q ≤ ∞, and norm if 1 ≤ p, q ≤ ∞. Moreover, (2) is independent of the choice of ϕ = {ϕ k } k∈Z n . We refer the reader to [5,6,8] for equivalent definitions (continuous versions).
We writex = (x 1 , x 2 , ..., x n−1 ) and define the anisotropic Wiener amalgam spaces W p,q,r by the following norm, Similarly, forx = (x 1 , x 2 , ..., x n−2 ), we define Comparing amalgam spaces W p,q s with anisotropic amalgam spaces W p,q,r s we see that W p,q s is, but W p,q,r s is not rotational invariant. Using the almost orthogonality of ϕ we see that the W p,q,r s is independent of ϕ. Moreover, recalling that ||f || W p,q,r s is the function sequence { k f } k∈Z n equipped with the L p ℓ r kn ℓ q k norm, it is easy to see that W p,q,r s is a quasi-Banach space for any s ∈ R, p, q, r ∈ (0, ∞] and a Banach space for any s ∈ R, 1 ≤ p, q, r ≤ ∞. Moreover, the Schwartz space is dense in W p,q,r s if p, q, r < ∞. The proofs are similar to those of amalgam spaces in [5,6,8].
We collect properties of Wiener amalgam spaces in the following lemma.
The last term is equivalent to here (s − s ′ )r = 1 + εr > 1 and s ′ ≥ 0 have been used.
(III -ii): Using the embedding ℓ q ֒→ ℓ r , In the last inequality, we need s ≥ 0.
Lemma 2.2 (Triebel, [14]). Let 0 < p < ∞ and 0 < q ≤ ∞. Let Ω = {Ω k } k∈Z n be a sequence of compact subsets of R n . Let d k be the diameter of Ω k . If 0 < r < min(p, q), then there exist a constant c such that . Then are equivalent norms in W p,q s (R n ) and W p,q,r s (R n ), respectively.
The proof is a direct consequence of Lemma 2.2, taking f k = k f . See also [13,Proposition].

Proof of the main results
First,we narrate the idea of the proof. We give an equivalent formulation for k(Tf )(x), a function in R n−1 , via some k ,l f (x, 0) a function in R n . Then we compute for pointwise estimates between the corresponding ℓ q norms and ℓ r kn ℓ q k norms for cases 0 < q < 1 and 1 ≤ q < ∞, separately. Finally, taking L p (R n−1 ) norms and using our equivalent norms in Proposition (2.1), we arrive to our conclusion.
We denote Fx(F −1 ξ ) the partial (inverse) Fourier transform onx (ξ) ∈ R n−1 . Write {ϕk}k ∈Z n−1 as versions of (1) in R n−1 . By the support property of ϕk, we observe where ψk ,l (ξ) = ϕk(ξ)ϕ l (ξ), l = (l, l n ), and k ,l f := F −1 ψk ,l Ff . Note that the left-hand side is a function in R n−1 while the right-hand side is a function in R n .
Recall our maximal function (3) and take y 1 = y 2 = · · · = y n−1 = 0, y n = x n we have for |x n |≤ 1, Proof of Theorem 1.1. We start by taking the ℓ q -norm of (6). We write,  For 0 < q < 1, we estimate (8) by where e j is the j th column of the identity matrix. In the sequel, it suffice to consider only the case j = 1. Moreover, we write l f := l±e 1 ,l f for some ψ l satisfying (1). Using (7) we have, Combining (9) and (10), then taking the L p (R n−1 )-norm and raising to p-th power gives, Integrating over x n ∈ [0, 1], Note that the last inequality follows from Proposition 2.1.
For 1 ≤ q ≤ ∞, we use Minkowski's inequality to give an upper bound of (8) as follows,  Repeating the arguments above on (11) gives us the estimate, Hence, we arrive to our desired estimates.
Thus, T −1 : W p,q s (R n−1 ) → W p,q,q∧1 s (R n ). As the end of this paper, we discuss the optimality of Corollary 1.1. We recall the counterexample given in [9]. For 1 < p, q < ∞, there exist a function which shows T : M p,q 1/q ′ (R n ) → M p,q 0 (R n−1 ).
Since M q,q = W q,q , we also have T : W q,q 1/q ′ (R n ) → W q,q 0 (R n−1 ). Hence, Corollary 1.1 is sharp for p = q, 1 < p, q < ∞ (refer to FIGURE 1). We now claim that it is also sharp for all 1 < p, q < ∞. Contrary to our claim, suppose s = 1/q ′ implies TW p,q s (R n ) = W p,q (R n−1 ). Then, by interpolation with the estimate for a point Q(p 1 , q 1 ) with s = 1/q ′ 1 , one would obtain an improvement for the segment connecting P (p, q) and Q(p 1 , q 1 ) (refer to FIGURE 2), which is not possible.