Almost diagonalization of $\tau$-pseudodifferential operators with symbols in Wiener amalgam and modulation spaces

TL;DR

This paper extends the almost-diagonalization results of pseudodifferential operators to a broader class of symbols in Wiener amalgam and modulation spaces, enabling new boundedness and algebra properties.

Contribution

It generalizes the almost-diagonalization of Weyl operators to all $ au$-pseudodifferential operators with symbols in weighted Wiener amalgam spaces.

Findings

Extended almost diagonalization to $ au$-pseudodifferential operators.

Established boundedness and algebra properties on Wiener amalgam and modulation spaces.

Applied results to symbols in the Sj"ostrand class and related spaces.

Abstract

In this paper we focus on the almost-diagonalization properties of $\tau$-pseudodifferential operators using techniques from time-frequency analysis. Our function spaces are modulation spaces and the special class of Wiener amalgam spaces arising by considering the action of the Fourier transform of modulation spaces. A particular example is provided by the Sj\"ostrand class, for which Gr\"ochenig exhibited the almost diagonalization of Weyl operators. We shall show that such result can be extended to any $\tau$-pseudodifferential operator, for $\tau \in [0,1]$, also with symbol in weighted Wiener amalgam spaces. As a consequence, we infer boundedness, algebra and Wiener properties for $\tau$-pseudodifferential operators on Wiener amalgam and modulation spaces.