Pseudo-differential operators in a Gelfand-Shilov setting

TL;DR

This paper introduces new classes of pseudodifferential operators with exponential growth symbols and studies their properties on Gelfand-Shilov spaces, including mapping, composition, and invariance, in both quasi-analytic and non-quasi-analytic cases.

Contribution

It develops a comprehensive framework for pseudodifferential operators with exponential growth symbols in Gelfand-Shilov spaces, including new composition and invariance results.

Findings

Established mapping properties on Gelfand-Shilov spaces

Proved composition theorems for these operators

Analyzed invariance properties in different cases

Abstract

We introduce some general classes of pseudodifferential operators with symbols admitting exponential type growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces both in the quasi-analytic and in the non-quasi-analytic case. Moreover, we prove composition theorems and certain invariance properties of these classes.