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 considers 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 Ces\`{a}ro 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 three lines lemma.