Orthonormal Strichartz inequalities for the $(k, a)$-generalized Laguerre operator and Dunkl operator

TL;DR

This paper establishes Strichartz inequalities for the $(k,a)$-generalized Laguerre and Dunkl operators, providing new restriction theorems and applications to Schrödinger equations with these operators.

Contribution

It introduces novel Strichartz estimates for orthonormal families related to the $(k,a)$-generalized Laguerre and Dunkl operators, extending classical results to these operators.

Findings

Proved a restriction theorem for the Fourier-$\

$\\Delta_{k,a}$ transform.

Established Strichartz estimates for Schrödinger propagators associated with these operators.

Abstract

Let $\Delta_{k,a}$ and $\Delta_k $ be the $(k,a)$-generalized Laguerre operator and the Dunkl Laplacian operator on $\mathbb{R}^n$, respectively. The aim of this article is twofold. First, we prove a restriction theorem for the Fourier-$\Delta_{k,a}$ transform. Next, as an application of the restriction problem, we establish Strichartz estimates for orthonormal families of initial data for the Schr\"odinger propagator $e^{-i t \Delta_{k, a}} $ associated with the operator $ \Delta_{k, a}$. Further, using the classical Strichartz estimates for the free Schr\"odinger propagator $e^{-i t \Delta_{k, a}} $ for orthonormal systems of initial data and the kernel relation between the semigroups $e^{-i t \Delta_{k, a}}$ and $e^{i \frac{t}{a}\|x\|^{2-a} \Delta_{k}},$ we prove Strichartz estimates for orthonormal systems of initial data associated with the Dunkl operator $ \Delta_k $ on $\mathbb{R}^n$. Finally, we present some applications to our aforementioned results.