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

We perform Wigner analysis of linear operators. Namely, the standard time frequency representation Short-time Fourier Transform (STFT) is replaced by the A-Wigner distribution defined by W-A(f) = mu(A)(f circle times f), where A is a 4d x 4d symplectic matrix and mu(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 is an element of (0, 1). Furthermore, they can be efficiently employed in the study of the off-diagonal decay for pseudodifferential operators with symbols in the Sjostrand class (in particular, in the Hormander 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 Hormander wave front set and identify the possible presence of a ghost region in the Wigner wave front. In the second part of the paper applications to Fourier integral operators and Schrodinger equations will be given (c) 2022 Elsevier Inc. All rights reserved.