Wigner analysis of operators. Part I: pseudodifferential operators and wave fronts

TL;DR

This paper introduces a novel Wigner analysis framework for linear operators using metaplectic transformations, offering new insights into pseudodifferential operators, modulation spaces, and wave front sets.

Contribution

It develops a new time-frequency representation via metaplectic operators, enhancing the understanding of quantization and microlocal analysis for pseudodifferential operators.

Findings

Characterizes modulation spaces using $ au$-Wigner distributions.

Analyzes off-diagonal decay of pseudodifferential operators in Sj"ostrand class.

Identifies relationships between Wigner and H"ormander wave front sets.

Abstract

We perform Wigner analysis of linear operators. Namely, the standard time-frequency representation \emph{Short-time Fourier Transform} (STFT) is replaced by the $\mathcal{A}$-\emph{Wigner distribution} defined by $W_{\mathcal A} (f)=\mu({\mathcal A})(f\otimes\bar{f})$, where ${\mathcal A}$ is a $4d\times 4d$ symplectic matrix and $\mu({\mathcal A})$ is an associate metaplectic operator. Basic examples are given by the so-called $\tau$-Wigner distributions. Such representations provide a new characterization for modulation spaces when $\tau\in (0,1)$. Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sj\"ostrand class (in particular, in the H\"{o}rmander class $S^0_{0,0}$). The novelty relies on defining time-frequency representations via metaplectic operators, developing a conceptual framework and paving the way for a new understanding of quantization procedures. We deduce micro-local properties for pseudodifferential operators in terms of the Wigner wave front set. Finally, we compare the Wigner with the global H\"{o}rmander wave front set and identify the possible presence of a ghost region in the Wigner wave front. \par In the second part of the paper applications to Fourier integral operators and Schr\"{o}dinger equations will be given.