Besov and Triebel-Lizorkin Spaces Associated to Hermite Operators

Consider the Hermite operator H=-Δ+|x|2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H=-\Delta +|x|^2$$\end{document} on the Euclidean space Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}. The main aim of this article is to develop a theory of homogeneous and inhomogeneous Besov and Triebel–Lizorkin spaces associated to the Hermite operator. Our inhomogeneous Besov and Triebel–Lizorkin spaces are different from those introduced by Petrushev and Xu (J Fourier Anal Appl 14, 372–414 2008). As applications, we show the boundedness of negative powers and spectral multipliers of the Hermite operators on some appropriate Besov and Triebel–Lizorkin spaces.