Fractional Leibniz rules for the Dunkl Laplacian in Besov and Triebel--Lizorkin spaces

Abstract

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(ν)\geq 0$, $ν\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on Besov and Triebel--Lizorkin spaces in the Dunkl setting. Our approach exploits the interplay between spectral multipliers and the Dunkl transform, together with the support properties of the distributions associated with Dunkl translations. These results extend the corresponding Leibniz-type estimates previously established on $L^p$ spaces to the broader setting of Besov and Triebel--Lizorkin spaces.