Dual of modulation spaces

The aim of this paper is the study of the dual of modulation spaces M(R) for 0 < p, q < ∞ .


Introduction
We have constructed the modulation spaces M p,q (R d ) in [2] for general 0 < p, q ∞, which coincide with the ususal modulation spaces when 1 p, q ∞, and studied their basic properties.The aim of this paper is the study of the dual of M p,q (R d ) for 0 < p, q < ∞.When 1 p, q < ∞, the fact that M p ,q (R d ) is the dual of M p,q (R d ) is already known, where 1/p + 1/p = 1/q + 1/q = 1 .(See Feichtinger [1].)So in this paper we are concerned with the dual, in particular when p < 1 or q < 1 .Motivated by the fact that the modulation spaces have similar properties to that of the Besov spaces (Proposition 2.2), we employ H. Triebel's method [3] to study the dual.But gained results are similar to the sequence spaces l p rather than the Besov spaces B s p,q (R d ).

Basic definition.
The following notations will be used throughout this article.Let S(R d ) be the Schwartz space of all complex-valued rapidly decreasing infinitely differentiable functions on R d and S (R d ) be the topological dual of S(R d ).The Fourier transform is f (ω) = f (t)e −2πiω•t dt, and the inverse Fourier transform is f (t) = f (−t).We define for 0 For a function f on R d , the translation and the modulation operators are defined by respectively.And f is the function defined by f (x) := f (−x).

Modulation spaces and basic properties.
We recall the definition of the modulation spaces.
First for α > 0 we define Φ α (R d ) to be the space of all g ∈ S(R d ) satisfying supp g ⊂ {ξ | |ξ| 1}, and In the following, we choose a sufficiently small α > 0 so that the function space Φ α (R d ) is not empty.With this, we have defined the modulation spaces as follows: Definition 2.1.Given a g ∈ Φ α (R d ), and 0 < p, q ∞, we define the modulation space M p,q (R d ) to be the space of all tempered distributions f ∈ S (R d ) such that the quasi-norm is finite, with obvious modifications if p or q = ∞.
We state basic properties of modulation spaces, which will play an important role in this article.(See [2].)Proposition 2.2.Let 0 < p, q ∞ and g ∈ Φ α (R d ).Then (a) (d) We have the continuous embeddings These facts have been derived from the following.Let 0 < p ∞, and Γ be a compact subset of R d .Then L p Γ is defined by Lemma 2.3.Let Γ be a compact subset of R d and let 0 < p q ∞.Then there exists a positive constant C (which depends only on the diameter of Γ and p) such that ||f || L q C||f || L p holds for all f ∈ L p Γ .Lemma 2.4.Let 0 < p 1 and Γ, Γ be compact subsets of R d .Then there exists a positive constant C (which depends only on the diameters of Γ, Γ and p ) such that Γ and all g ∈ L p Γ .In the sequel, we shall not distinguish between equivalent quasi-norms of a given quasi-normed space.

Summary of the results.
We now formulate our results.
(i) For 0 < p, q 1, we have Here the symbol M p,q (R d ) is used to denote all bounded linear functionals on M p,q (R d ) and p is a conjugate exponent of p which is determined in the usual way by 1  p + 1 p = 1 if 1 p < ∞, and we put p = ∞ if 0 < p < 1. q is determined in the same way.Suppose 0 < p, q < 1 .Then for the sequence spaces, (l p ) = l ∞ , that is all dual spaces are equal.But for the Besov spaces, Triebel [3] . In this sense, the modulation spaces are similar to the sequence spaces rather than the Besov spaces for the dual.At present, we don't know whether or not M p,q (R d ) = M p ,q (R d ) when 0 < p < 1 < q < ∞.

Technical lemmas. Our theory deeply depends on the following transformation from a function
It is natural to ask whether it is possible to recover f .The following key lemma in this paper asserts that this transformation is invertible on S (R d ) and also implies the duality on M p,q (R d ).

Lemma 2.5. (The Inversion Formula
Then there exists a positive constant N which depends only on the size of supp ĝ , α > 0 and the dimension d, such that Proof.First note that there exists a constant N (depending only on the size of supp g , α > 0 and dimension d) such that Then the convolution f * η exists in O M and can be represented in the form

Dual spaces
In this section we study the space M p,q (R d ) * of bounded anti-linear functionals on M p,q (R d ), equivalently, study the dual of M p,q (R d ).We mean here, a functional l on a quasi-Banach space B is anti-linear if for any φ, ψ ∈ B and α ∈ C, we have l(φ+ψ) = l(φ)+l(ψ) and l(αφ) = ᾱl(φ).Throughout this section, g denotes a function in Φ α (R d ).If 1 p < ∞, then the conjugate exponent p of p is determined in the usual way by Then the anti-linear functional l given by initially defined on the dense subspace S(R d ) of M p,q (R d ), has a unique bounded extension to M p,q (R d ) and satisfies (b) Conversely, every bounded anti-linear functional l on M p,q (R d ) can be realized as above, with an f ∈ M p ,∞ (R d ), and with Then by Hölder's inequality and continuous embedding for sequence spaces, l q ⊂ l 1 if q 1, we have Thus l(ψ) defined by (1) converges and defines a bounded anti-linear functional l on is a dense subset of M p,q (R d ).Therefore, setting l, ψ := l(ψ), we obtain (1) with f = l by virtue of Lemma 2.5.Let 1 p < ∞, 0 < q 1 and l ∈ M p,q (R d ) * .We show that for By Lemma 2.7, the following relation holds true: From this and Young's inequality, we have Taking supremum over k ∈ Z d , we obtain the estimate 3.2.The case 0 < p < 1.
Theorem 3.2.Let 0 < p < 1 and 0 < q < ∞.(a) Suppose f ∈ M ∞,q (R d ).Then the anti-linear functional l given by initially defined on the dense subspace S(R d ) of M p,q (R d ), has a unique bounded extension to M p,q (R d ) and satisfies (b) Every bounded anti-linear functional l on M p,q (R d ) can be realized as above, with an f ∈ M ∞,∞ (R d ), and with Proof.(a).Let f ∈ M ∞,q (R d ).Then using Lemma 2.3, we have C||f || M ∞,q ||ψ|| M p,q .
In the last inequality, we use Hölder's inequality with k ∈ Z d if q 1, and continuous embedding l q ⊂ l 1 if q 1. (b).Let l ∈ M p,q (R d ) * .Then using the definition of a convolution, we holds for all x 0 ∈ R d and k ∈ Z d .Moreover, It follows from by Lemma 2.4 that the last sum is bounded by where C is independent of x 0 and k .Combining this estimate with (3) and taking supremum over x 0 ∈ R d and k ∈ Z d , we get the desired result.