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

TL;DR

This paper establishes Strichartz estimates for a broad class of generalized Laguerre operators involving Dunkl Laplacians, extending previous results for specific cases and employing advanced harmonic analysis techniques.

Contribution

It generalizes Strichartz estimates to the $(k,a)$-generalized Laguerre operators for a wider range of parameters, using novel symbol estimates and stationary phase methods.

Findings

Proved Strichartz estimates for generalized Laguerre operators with Dunkl Laplacians.

Extended previous results from specific cases to more general parameters.

Developed new analytical techniques involving symbol estimates and discrete stationary phase.

Abstract

In this paper, we prove Strichartz estimates for the $(k,a)$-generalized Laguerre operators $a^{-1}\bigl(-|x|^{2-a}\Delta_k+|x|^a\bigr)$ which were introduced by Ben Sa\"{\i}d-Kobayashi-Orsted, and for the operators $|x|^{2-a}\Delta_k$. Here $k$ denotes a non-negative multiplicity function for the Dunkl Laplacian $\Delta_k$ 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.