Anisotropic global microlocal analysis for tempered distributions

TL;DR

This paper develops an anisotropic extension of the Shubin pseudodifferential calculus, introducing anisotropic symbols and Gabor wave front sets, and explores their microlocal properties and applications to oscillatory functions.

Contribution

It introduces anisotropic symbols and Gabor wave front sets within the Shubin calculus, providing new microlocal analysis tools for phase space analysis.

Findings

Defined anisotropic symbols and Gabor wave front sets.

Established subcalculi of the isotropic Shubin calculus.

Proved inclusion properties for anisotropic Gabor wave front sets of oscillatory functions.

Abstract

We study an anisotropic version of the Shubin calculus of pseudodifferential operators on $\mathbf R^d$. Anisotropic symbols and Gabor wave front sets are defined in terms of decay or growth along curves in phase space of power type parametrized by one positive parameter that distinguishes space and frequency variables. We show that this gives subcalculi of Shubin's isotropic calculus, and we show a microlocal as well as a microelliptic inclusion in the framework. Finally we prove an inclusion for the anisotropic Gabor wave front set of chirp type oscillatory functions with a real polynomial phase function.