Norm Estimates for $\tau$-Pseudodifferential Operators in Wiener Amalgam and Modulation Spaces

TL;DR

This paper investigates the boundedness of $ au$-pseudodifferential operators on modulation spaces, establishing uniform bounds for $ au$ in (0,1) and analyzing unboundedness at the endpoints, with implications for time-frequency analysis.

Contribution

It provides a uniform upper bound for the operator norm of $ au$-pseudodifferential operators across $ au in [0,1]$, extending known continuity results.

Findings

Boundedness for $ au in ext{endpoints}$

Unboundedness at $ au=0$ and $ au=1$

Uniform operator norm bound independent of $ au$

Abstract

We study continuity properties on modulation spaces for $\tau$-pseudodifferential operators with symbols $a$ in Wiener amalgam spaces. We obtain boundedness results for $\tau \in (0,1)$ whereas, in the end-points $\tau=0$ and $\tau=1$, the corresponding operators are in general unbounded. Furthermore, for $\tau \in (0,1)$, we exhibit a function of $\tau$ which is an upper bound for the operator norm. The continuity properties of $\tau$-pseudodifferential operators, for any $\tau\in [0,1]$, with symbols $a$ in modulation spaces are well known. Here we find an upper bound for the operator norm which does not depend on the parameter $\tau \in [0,1]$, as expected. Key ingredients are uniform continuity estimates for $\tau$-Wigner distributions.