Mixed norm estimates for the Cesàro means associated with DunklHermite expansions
Abstract
Our main goal in this article is to study mixed norm estimates for the Cesàro means associated with DunklHermite expansions on $\mathbb{R}^d$. These expansions arise when one consider the DunklHermite operator (or Dunkl harmonic oscillator) $H_{\kappa}:=\Delta_{\kappa}+x^2$, where $\Delta_{\kappa}$ stands for the DunklLaplacian. It is shown that the desired mixed norm estimates are equivalent to vectorvalued 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.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.2162
 Bibcode:
 2014arXiv1410.2162B
 Keywords:

 Mathematics  Classical Analysis and ODEs;
 Mathematics  Functional Analysis
 EPrint:
 24 pages. Revised version following referee's comments. To appear in Transactions of the American Mathematical Society