Besov and Triebel-Lizorkin spaces associated with Laguerre expansions of Hermite type

Abstract

Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $\alpha \in [-1/2,\infty {)}^d\backslash (-1/2,1/2{)}^d$, $d\ge 1$, are defined and investigated. To achieve expected goals Schwartz type spaces on $R_{+}^d$ are introduced and then tempered type distributions are constructed. Also, ideas from a recent paper of Bui and Duong on Besov and Triebel-Lizorkin spaces associated with Hermite functions expansions are used. This means, in particular, using molecular decomposition and an appropriate form of a Calderón reproducing formula.