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

Abstract

Let $\hat{I}{”}_{k,a}$ and $\hat{I}{”}_k$ 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-$\hat{I}{”}_{k,a}$ transform. Next, as an application of the restriction problem, we establish Strichartz estimates for orthonormal families of initial data for the Schrödinger propagator $e^{-it\hat{I}{”}_{k,a}}$ associated with the operator $\hat{I}{”}_{k,a}$. Further, using the classical Strichartz estimates for the free Schrödinger propagator $e^{-it\hat{I}{”}_{k,a}}$ for orthonormal systems of initial data and the kernel relation between the semigroups $e^{-it\hat{I}{”}_{k,a}}$ and $e^{i\frac{t}{a}\Vert x{\Vert }^{2-a}\hat{I}{”}_k},$ we prove Strichartz estimates for orthonormal systems of initial data associated with the Dunkl operator $\hat{I}{”}_k$ on ${\mathbb{R}}^n$. Finally, we present some applications to our aforementioned results.

24pages. All comments are welcome