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

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

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