Anti-Wick and Weyl quantization on ultradistribution spaces

TL;DR

This paper explores the relationship between Anti-Wick and Weyl quantization on ultradistribution spaces, identifying the largest subspace where Gaussian convolution exists, thus extending quantization to broader symbol classes.

Contribution

It establishes the connection between Anti-Wick and Weyl quantization for ultradistributions and extends the quantization to non-tempered ultradistribution symbols.

Findings

Identified the largest subspace of ultradistributions with Gaussian convolution

Extended Anti-Wick quantization to broader symbol classes

Analyzed continuity of pseudodifferential operators on ultradistribution spaces

Abstract

The connection between the Anti-Wick and Weyl quantization is given for certain class of global symbols, which corresponding pseudodifferential operators act continuously on the space of tempered ultradistributions of Beurling, respectively, of Roumieu type. The largest subspace of ultradistributions is found for which the convolution with the gaussian kernel exist. This gives a way to extend the definition of Anti-Wick quantization for symbols that are not necessarily tempered ultradistributions.