On the hypercontractive property of the Dunkl-Ornstein-Uhlenbeck semigroup

Abstract

The aim of this paper is to prove the hypercontractive propertie of the Dunkl-Ornstein-Uhlenbeck semigroup, {e^(tL_k)}t≥0. To this end, we prove that the Dunkl-Ornstein-Uhlenbeck differential operator L_k with k ≥ 0 and associated to the ℤ^d₂ group, satisfies a curvature-dimension inequality, to be precise, a C(Ï, ∞)-inequality, with 0 ≤ Ï ≤ 1. As an application of this fact, we get a version of Meyer‘s multipliers theorem and by means of this theorem and fractional derivatives, we obtain a characterization of Dunkl-potential spaces.