Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

TL;DR

This paper proves fractional Leibniz rules for the Dunkl Laplacian using adapted paraproduct decompositions, introducing new auxiliary results like decay estimates and boundedness of Dunkl paraproducts.

Contribution

It extends classical fractional Leibniz rules to the Dunkl setting by developing Dunkl-specific paraproduct techniques and auxiliary estimates.

Findings

Established fractional Leibniz rules for the Dunkl Laplacian.

Developed decay estimates for $(- abla_k)^s f$ for Schwartz functions.

Proved boundedness of Dunkl paraproduct operators on Lebesgue spaces.

Abstract

We establish fractional Leibniz rules for the Dunkl Laplacian $\Delta_k$ of the form $$\|(-\Delta_k)^s(fg)\|_{L^p(d\mu_k)} \lesssim \|(-\Delta_k)^s f\|_{L^{p_1}(d\mu_k)} \|g\|_{L^{p_2}(d\mu_k)} + \|f\|_{L^{p_1}(d\mu_k)} \|(-\Delta_k)^s g\|_{L^{p_2}(d\mu_k)}.$$ Our approach relies on adapting the classical paraproduct decomposition to the Dunkl setting. In the process, we develop several new auxiliary results. Specifically, we show that for a Schwartz function $f$, the function $(-\Delta_k)^s f$ satisfies a pointwise decay estimate; we establish a version of almost orthogonality estimates adapted to the Dunkl framework; and we investigate the boundedness of Dunkl paraproduct operators on the Lebesgue spaces.