Dunkl paraproducts and fractional Leibniz rules for the Dunkl Laplacian

We establish fractional Leibniz rules for the Dunkl Laplacian ${\mathrm{\Delta }}_k$ of the form $$\mathrm{\parallel }(-{\mathrm{\Delta }}_k{)}^s(fg){\mathrm{\parallel }}_{L^p(d{\mu }_k)}\lesssim \mathrm{\parallel }(-{\mathrm{\Delta }}_k{)}^{s}f{\mathrm{\parallel }}_{L^{p_1}(d{\mu }_k)}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^{p_2}(d{\mu }_k)}+\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{p_1}(d{\mu }_k)}\mathrm{\parallel }(-{\mathrm{\Delta }}_k{)}^{s}g{\mathrm{\parallel }}_{L^{p_2}(d{\mu }_k)}\mathrm{.}$$ 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 $(-{\mathrm{\Delta }}_k{)}^{s}f$ 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.