Abstract Let $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Rd which contains at least one non-zero element the Feichtinger algebra $$S_0$$ xmlns:mml=...

The α-modulation spaces M p,q , introduced by Gröbner in [10] are a class of function spaces that contain Besov spaces B p,q (α = 1) and modulation spaces M s p,q (α = 0) as special cases. There are two kinds of basic coverings on Euclidean R which is very useful in the theory of function spaces and their applications, one is the uniform covering R = ⋃ k∈Zn Qk, where Qk denote the unit cube wit...

This is a “problems” paper. We isolate some connections between operator theory and the theory of modulation spaces that were stimulated by a question of Feichtinger’s regarding integral and pseudodifferential operators. We discuss several problems inspired by this question, and give a reformulation of the original question in operator-theoretic terms. A detailed discussion of the background an...

We give a complete characterization of the continuity of pseudodifferential operators with symbols in modulation spaces M, acting on a given Lebesgue space L. Namely, we find the full range of triples (p, q, r), for which such a boundedness occurs. More generally, we completely characterize the same problem for operators acting on Wiener amalgam space W (L, L) and even on modulation spaces M . ...

Abstract We deduce continuity properties for pseudo-differential operators with symbols in quasi-Banach Orlicz modulation spaces when rely on other spaces. In particular we extend some earlier results.

We consider the dilation property of the modulation spaces M. Let Dλ : f(t) 7→ f(λt) be the dilation operator, and we consider the behavior of the operator norm ‖Dλ‖Mp,q→Mp,q with respect to λ. Our result determines the best order for it, and as an application, we establish the optimality of the inclusion relation between the modulation spaces and Besov space, which was proved by Toft [9].

In this paper we sharpen a theorem of Gröchenig about embedding weighted Lp-spaces into a modulation space. The theorem is closely related to the problem of pointwise convergence in the Poisson summation formula (PSF). We also show that this sharp version is optimal in some sense by extending a family of counterexamples to PSF also due to Gröchenig. The proof requires new techniques, namely the...