Hörmander's multiplier theorem for the Dunkl transform

Abstract

For a normalized root system R in RN and a multiplicity function k≥0 let N=N+∑α∈Rk(α). Denote by dw(x)=∏α∈R|〈x,α〉|k(α)dx the associated measure in RN. Let F stand for the Dunkl transform. Given a bounded function m on RN, we prove that if there is s>N such that m satisfies the classical Hörmander condition with the smoothness s, then the multiplier operator Tmf=F−1(mFf) is of weak type (1,1), strong type (p,p) for 1<p<∞, and is bounded on a relevant Hardy space H1. To this end we study the Dunkl translations and the Dunkl convolution operators and prove that if F is sufficiently regular, for example its certain Schwartz class seminorm is finite, then the Dunkl convolution operator with the function F is bounded on Lp(dw) for 1≤p≤∞. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.