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

In this paper we find new equivalent norms in Lp(Rn,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p}({\mathbb {R}}^n,{\mathbb {B}})$$\end{document} by using multivariate Littlewood–Paley functions associated with Poisson semigroup for the Hermite operator, provided that B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {B}}$$\end{document} is a UMD Banach space with the property (α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}). We make use of γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma $$\end{document}-radonifying operators to get new equivalent norms that allow us to obtain Lp(Rn,B)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p({\mathbb {R}}^n,{\mathbb {B}})$$\end{document}-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.