Mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions

Abstract

Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with Dunkl--Hermite expansions on ${\mathbb{R}}^d$. These expansions arise when one consider the Dunkl--Hermite operator (or Dunkl harmonic oscillator) $H_{\kappa }:=-{\Delta }_{\kappa }+|x{|}^2$, 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 Cesàro means for Laguerre expansions with shifted parameter. In order to obtain the latter, we develop an argument to extend these operators for complex values of the parameters involved and apply a version of three lines lemma.

24 pages. Revised version following referee's comments. To appear in Transactions of the American Mathematical Society

Cited by in corpus (2)

• On Harmonic Analysis Operators in Laguerre-Dunkl and Laguerre-Symmetrized Settings
• Hardy-type inequalities for fractional powers of the Dunkl--Hermite operator