Narrow Results

The purpose of this paper is to investigate the mapping properties of the strongly singular convolution operators on general weighted modulation spaces for 0 < p ≤ ∞, 0 < q ≤ ∞ and s ∈ ℝ. Our results show that modulation spaces are good substitutions for Lebesgue spaces.

We study a class of almost diagonal matrices compatible with the mixed-norm $\alpha$-modulation spaces $M_{p,q}^{s,\alpha }({ℝ}^n)$, $\alpha \in [0,1]$, introduced recently by Cleanthous and Georgiadis. The class of almost diagonal matrices is shown to be closed under matrix multiplication and we connect the theory to the continuous case by identifying a suitable notion of molecules for the mixed-norm $\alpha$-modulation spaces. It is shown that the “change of frame” matrix for a pair of time-frequency frames for the mixed-norm $\alpha$-modulation consisting of suitable molecules is almost diagonal. As examples of applications, we use the almost diagonal matrices to construct compactly supported frames for the mixed-norm $\alpha$-modulation spaces, and to obtain a straightforward boundedness result for Fourier multipliers on the mixed-norm $\alpha$-modulation spaces.

Localization operators have been object of study in quantum mechanics, in PDE and signal analysis recently. In engineering, a natural language is given by time-frequency analysis. Arguing from this point of view, we shall present the theory of these operators developed so far. Namely, regularity properties, composition formulae and their multilinear extension shall be highlighted. Time-frequency analysis will provide tools, techniques and function spaces. In particular, we shall use modulation spaces, which allow “optimal” results in terms of regularity properties for localization operators acting on L²(ℝ^d).