Hörmander's multiplier theorem for the Dunkl transform

Abstract For a normalized root system R in R N and a multiplicity function k ≥ 0 let N = N + ∑ α ∈ R k ( α ) . Denote by d w ( x ) = ∏ α ∈ R | 〈 x , α 〉 | k ( α ) d x the associated measure in R N . Let F 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 ( m F f ) is of weak type ( 1 , 1 ) , strong type ( p , p ) for 1 p ∞ , 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 ( d w ) for 1 ≤ p ≤ ∞ . We also consider boundedness of maximal operators associated with the Dunkl convolutions with Schwartz class functions.