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

Boggarapu, Pradeep; Roncal, Luz; Thangavelu, Sundaram

doi:10.1090/tran/6861

Mixed norm estimates for the Cesàro means associated with Dunkl–Hermite expansions
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by Pradeep Boggarapu, Luz Roncal and Sundaram Thangavelu PDF

Trans. Amer. Math. Soc. 369 (2017), 7021-7047 Request permission

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 Cesàro 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.

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