We consider Fourier multipliers for L p associated with the Dunkl operator on R and establish a version of Ho¨rmander's multiplier theorem. In applying this version, we come up with some results regarding the oscillating multipliers, partial sum operators and generalized Bessel potentials. r 2004 Elsevier Inc. All rights reserved. MSC: primary 43A62; secondary 43A15; 43A32

In this paper we study a class of generalized Fock spaces associated with the Dunkl operators. Next we introduce the commutator relations between the Dunkl operators and multiplication operators which lead to a generalized class of Weyl commutation relations for the Dunkl kernel.

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Abstract. We consider the Dunkl intertwining operator V α and its dual t V α , we define and study the Dunkl Sonine operator and its dual on R. Next, we introduce complex powers of the Dunkl Laplacian ∆ α and establish inversion formulas for the Dunkl Sonine operator S α,β and its dual t S α,β . Also, we give a Plancherel formula for the operator t S α,β .