Hormander's multiplier theorem for the Dunkl transform

摘要

For a normalized root system R in R-N and a multiplicity function k >= 0 let N = N + Sigma(alpha is an element of R)k(alpha). Denote by dw(x) = Pi(alpha is an element of R) vertical bar < x, alpha >vertical bar(k(alpha)) dx the associated measure in RN. Let stand for the Dunkl transform. Given a bounded function m on R-N, we prove that if there is s > N such that m satisfies the classical Hormander condition with the smoothness s, then the multiplier operator T(m)f = F-1(mFf) is of weak type (1, 1), strong type (p,p) for 1 < p < infinity, and is bounded on a relevant Hardy space H-1. 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 L-p(dw) for 1 <= p <= infinity. We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions. (C) 2019 Elsevier Inc. All rights reserved.