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

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.