Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

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.