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

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