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

TL;DR

This paper establishes new Strichartz estimates for systems of orthonormal functions related to the special Hermite operator, extending previous results and analyzing endpoint cases and spectral projections in Schatten spaces.

Contribution

It introduces novel restriction estimates for the Fourier-special Hermite transform and generalizes spectral projection estimates within trace ideals.

Findings

New Strichartz estimates for orthonormal functions associated with the special Hermite operator.

Extension of restriction estimates to spectral projections in Schatten spaces.

Analysis of endpoint cases for the Schrödinger propagator.

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 $e^{-it\phi(\mathcal{L})}$, where $ \phi : \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 $\{(\lambda, \mu, \nu)\in \mathbb{R}\times\mathbb{N}_0^n\times\mathbb{N}_0^n : \lambda=\phi(2|\nu|+n)\}$. We also discuss the endpoint case of the orthonormal Strichartz estimate for the Schr\"{o}dinger propagator $e^{-it\mathcal{L}}$. Furthermore, we generalize restriction estimates for the special Hermite spectral projections in the context of trace ideals (Schatten spaces).