Restriction theorem for Fourier-Dunkl transform II: Paraboloid, sphere, and hyperboloid surfaces

TL;DR

This paper extends the Fourier-Dunkl transform restriction theorem to paraboloid, sphere, and hyperboloid surfaces, and applies these results to derive Strichartz estimates for Schrödinger and Klein-Gordon equations involving Dunkl operators.

Contribution

It proves restriction theorems for Fourier-Dunkl transform on new surfaces and generalizes to orthonormal functions, with applications to PDE Strichartz estimates.

Findings

Restriction theorem established for paraboloid, sphere, and hyperboloid surfaces.

Strichartz estimates derived for Dunkl Schrödinger and Klein-Gordon equations.

Extension to orthonormal families of initial data.

Abstract

This is a continuation of the paper "Restriction theorem for Fourier-Dunkl transform I: Cone surface, J. Pseudo-Differ. Oper. Appl. 14(1), Paper No. 5 (2023)", where the authors introduced and studied the Fourier-Dunkl transform on $\mathbb{R}^{n}\times\mathbb{R}^{d}$. The main novelty of this paper is that we here prove Strichartz's restriction theorem for the Fourier-Dunkl transform for certain surfaces, namely, paraboloid, sphere, and hyperboloid and its generalisation to the family of orthonormal functions. Finally, as an application of these restriction theorems, we establish versions of Strichartz estimates for orthonormal families of initial data associated with Schr\"odinger's propagator in the case of the Dunkl Laplacian and Klein-Gordon operator.