On the $L^2$ boundedness of pseudo-multipliers for Hermite expansions

TL;DR

This paper establishes conditions under which Hermite pseudo-multipliers are bounded on L^2 spaces, introduces new results for Gaussian measures, and develops a novel integration by-parts formula for Hermite expansions.

Contribution

It provides new boundedness criteria for Hermite pseudo-multipliers on L^2 and extends results to Gaussian measure settings with a novel integration by-parts approach.

Findings

Hermite pseudo-multipliers are bounded on L^2 under new conditions

Results extend to L^p spaces and Gaussian measures

Introduces a new integration by-parts formula for Hermite expansions

Abstract

We give various conditions for Hermite pseudo-multipliers to be bounded on $L^2(\mathbb{R}^n)$. As a by-product we also give results on $L^p(\mathbb{R}^n)$, as well as new results for pseudo-multipliers for the Gaussian measure setting. One of our key tools is a new integration by-parts formula for Hermite expansions.