Microlocal analysis for Gelfand--Shilov spaces

TL;DR

This paper develops a microlocal analysis framework for Gelfand--Shilov ultradistributions, introducing an anisotropic wave front set that captures phase space oscillations and decay properties, with applications to pseudodifferential operators.

Contribution

It introduces a new anisotropic wave front set for Gelfand--Shilov spaces and establishes microlocal results for pseudodifferential operators acting on these spaces.

Findings

Defined an anisotropic global wave front set for Gelfand--Shilov ultradistributions.

Proved microlocal results for pseudodifferential operators with specific symbol classes.

Determined wave front sets of derivatives of delta functions and exponential functions.

Abstract

We introduce an anisotropic global wave front set of Gelfand--Shilov ultradistributions with different indices for regularity and decay at infinity. The concept is defined by the lack of super-exponential decay along power type curves in the phase space of the short-time Fourier transform. This wave front set captures the phase space behaviour of oscillations of power monomial type, a k a chirp signals. A microlocal result is proved with respect to pseudodifferential operators with symbol classes that give rise to continuous operators on Gelfand--Shilov spaces. We determine the wave front set of certain series of derivatives of the Dirac delta, and exponential functions.