Quantizations and global hypoellipticity for pseudodifferential operators of infinite order in classes of ultradifferentiable functions

TL;DR

This paper investigates the effects of different quantizations on infinite order pseudodifferential operators within ultradifferentiable function classes, analyzing composition, transpose, and hypoellipticity properties, with applications to Weyl calculus.

Contribution

It introduces a detailed analysis of quantization changes and hypoellipticity for infinite order pseudodifferential operators in ultradifferentiable settings, extending existing theories.

Findings

Comparison of different quantizations and their compositions

Analysis of transpose operations in the context of ultradifferentiable functions

Relations between global ω-hypoellipticity and regularity

Abstract

We study the change of quantization for a class of global pseudodifferential operators of infinite order in the setting of ultradifferentiable functions of Beurling type. The composition of different quantizations as well as the transpose of a quantization are also analysed, with applications to the Weyl calculus. We also compare global $\omega$-hypoellipticity and global $\omega$-regularity of these classes of pseudodifferential operators.