Sharp Continuity Results for the Short-Time Fourier Transform and for Localization Operators

TL;DR

This paper provides a complete characterization of the boundedness of the short-time Fourier transform and localization operators on various function spaces, establishing sharp and optimal conditions for their boundedness.

Contribution

It introduces new techniques to relax existing boundedness conditions and proves the optimality of these conditions for both $L^p$ and Wiener amalgam spaces.

Findings

STFT boundedness on $L^p$ spaces is sharp.

Sufficient conditions for boundedness on Wiener amalgam spaces are established and shown to be sharp.

Localization operators' boundedness conditions are relaxed and proven optimal.

Abstract

We completely characterize the boundedness on $L^p$ spaces and on Wiener amalgam spaces of the short-time Fourier transform (STFT) and of a special class of pseudodifferential operators, called localization operators. Precisely, a well-known STFT boundedness result on $L^p$ spaces is proved to be sharp. Then, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces $W(L^p,L^q)$ are given and their sharpness is shown. Localization operators are treated similarly. Using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for localization operators on $L^p$ spaces and prove the optimality of our results. More generally, we prove sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.