Restriction Theorem and Strichartz estimate for orthonormal functions associated with the Special Hermite Operator

Abstract

Let $\mathcal{L}$ be the special Hermite operator on ${\mathbb{C}}^n$. As a continuation of the recent results in \cite{SG}, we establish new Strichartz estimates for systems of orthonormal functions associated with general flows of the form $e^{-it\ddot{I}•(\mathcal{L})}$, where $\ddot{I}•:{\mathbb{R}}^{+}\to \mathbb{R}$ is a smooth function. Our approach relies on restriction estimates for the Fourier-special Hermite transform on the class of surfaces $\{(\hat{I}»,\hat{I}¼,\hat{I}½)\in \mathbb{R}\times {\mathbb{N}}_0^n\times {\mathbb{N}}_0^n:\hat{I}»=\ddot{I}•(2|\hat{I}½|+n)\}$. We also discuss the endpoint case of the orthonormal Strichartz estimate for the Schrödinger propagator $e^{-it\mathcal{L}}$. Furthermore, we generalize restriction estimates for the special Hermite spectral projections in the context of trace ideals (Schatten spaces).