Riesz Potentials, Bessel Potentials and Fractional Derivatives on Triebel-Lizorkin spaces for the Gaussian Measure

TL;DR

This paper extends the analysis of Riesz, Bessel potentials, and fractional derivatives to Gaussian Triebel-Lizorkin spaces, broadening the understanding of their boundedness properties and potential applications to other expansion frameworks.

Contribution

It proves the boundedness of these operators on Gaussian Triebel-Lizorkin spaces, extending previous results from Besov-Lipschitz spaces and applicable to broader diffusion semigroup contexts.

Findings

Boundedness of operators on Gaussian Triebel-Lizorkin spaces established

Results extend to Laguerre and Jacobi expansions

Framework applicable to general diffusion semigroups

Abstract

In a previous paper the boundedness properties of Riesz Potentials, Bessel potentials and Fractional Derivatives were studied in detail on Gaussian Besov-Lipschitz spaces $B_{p,q}^{\alpha}(\gamma_d)$. In this paper we will continue our study proving the boundedness of those operators on Gaussian Triebel-Lizorkin spaces $F_{p,q}^{\alpha}(\gamma_d)$. Also these results can be extended to the case of Laguerre or Jacobi expansions and even further to the general framework of diffusions semigroups.