Strichartz Estimates for the $(k,a)$-Generalized Laguerre Operators

Abstract

In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}(-|x{|}^{2-a}\hat{I}{”}_k+|x{|}^a)$ which were introduced by Ben Sa\"ıd-Kobayashi-Orsted, and for the operators $|x{|}^{2-a}\hat{I}{”}_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $\hat{I}{”}_k$ and $a$ denotes a positive real number satisfying certain conditions. The cases $a=1,2$ were studied previously. We consider more general cases here. The proof depends on symbol-type estimates of special functions and a discrete analog of the stationary phase theorem inspired by the work of Ionescu-Jerison.