Vector valued multivariate spectral multipliers, Littlewood-Paley functions, and Sobolev spaces in hte Hermite setting

TL;DR

This paper develops new equivalent norms in vector-valued $L^p$ spaces using multivariate Littlewood-Paley functions related to the Hermite operator, enabling boundedness results for spectral multipliers and showing the equivalence of Hermite Sobolev and potential spaces.

Contribution

It introduces novel norms in vector-valued $L^p$ spaces via Littlewood-Paley functions, establishing boundedness of spectral multipliers in the Hermite setting.

Findings

New equivalent norms in $L^p(R^n,B)$ using Littlewood-Paley functions.

Boundedness of vector-valued spectral multipliers for Hermite operators.

Hermite Sobolev and potential spaces are shown to coincide.

Abstract

In this paper we find new equivalent norms in $L^p(\mathbb{R}^n,\mathbb{B})$ by using multivariate Littlewood-Paley functions associated with Poisson semigroup for the Hermite operator, provided that $\mathbb{B}$ is a UMD Banach space with the property ($\alpha$). We make use of $\gamma$-radonifying operators to get new equivalent norms that allow us to obtain $L^p(\mathbb{R}^n,\mathbb{B})$-boundedness properties for (vector valued) multivariate spectral multipliers for Hermite operators. As application of this Hermite multiplier theorem we prove that the Banach valued Hermite Sobolev and potential spaces coincide.