Further results for the Dunkl Transform and the generalized Ces\`aro operator

Abstract

In this paper, we consider Dunkl theory on R^d associated to a finite reflection group. This theory generalizes classical Fourier anal- ysis. First, we give for 1 < p <= 2, sufficient conditions for weighted Lp-estimates of the Dunkl transform of a function f using respectively the modulus of continuity of f in the radial case and the convolution for f in the general case. In particular, we obtain as application, the integrability of this transform on Besov-Lipschitz spaces. Second, we provide necessary and sufficient conditions on nonnegative functions phi defined on [0; 1] to ensure the boundedness of the generalized Ces\`aro operator C_phi on Herz spaces and we obtain the corresponding operator norm inequalities.