Vector valued multivariate spectral multipliers, Littlewood–Paley functions, and Sobolev spaces in the Hermite setting

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.