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

TL;DR

This paper defines and studies Besov and Triebel-Lizorkin spaces linked to Laguerre function expansions of Hermite type, utilizing molecular decomposition and Calderón reproducing formulas for a comprehensive analysis.

Contribution

It introduces new Besov and Triebel-Lizorkin spaces associated with Laguerre expansions of Hermite type and develops their foundational properties using advanced harmonic analysis tools.

Findings

Spaces are rigorously defined and characterized.

Molecular decomposition techniques are established.

Connections to Schwartz spaces and tempered distributions are clarified.

Abstract

Homogeneous Besov and Triebel-Lizorkin spaces associated with multi-dimensional Laguerre function expansions of Hermite type with index $\alpha \in [-1/2,\infty)^d\backslash (-1/2,1/2)^d$, $d\geq 1$, 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 Calder\'on reproducing formula.