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 $$\Vert (-{\Delta }_k{)}^s(fg){\Vert }_{L^p(d{\mu }_k)}\lesssim \Vert (-{\Delta }_k{)}^{s}f{\Vert }_{L^{p_1}(d{\mu }_k)}\Vert g{\Vert }_{L^{p_2}(d{\mu }_k)}+\Vert f{\Vert }_{L^{p_1}(d{\mu }_k)}\Vert (-{\Delta }_k{)}^{s}g{\Vert }_{L^{p_2}(d{\mu }_k)}.$$ 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.

26 pages

References in corpus (1)

• Fractional Leibniz rule on the torus