MIXED NORM ESTIMATES FOR THE CESARO MEANS ASSOCIATED WITH DUNKL-HERMITE EXPANSIONS

Our main goal in this article is to study mixed norm estimates for the Cesaro means associated with Dunkl-Hermite expansions on R-d. These expansions arise when one considers the Dunkl-Hermite operator (or Dunkl harmonic oscillator) H-kappa := -Delta(kappa)+vertical bar x vertical bar(2), where Delta(kappa)stands for the Dunkl-Laplacian. It is shown that the desired mixed norm estimates are equivalent to vector-valued inequalities for a sequence of Cesaro means for Laguerre expansions with shifted parameter. In order to obtain such vector-valued inequalities, we develop an argument to extend these Laguerre operators for complex values of the parameters involved and apply a version of the three lines lemma.