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 $e^{-itφ(\mathcal{L})}$, where $ φ: \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 ${(λ, μ, ν)\in \mathbb{R}\times\mathbb{N}_0^n\times\mathbb{N}_0^n : λ=φ(2|ν|+n)}$. 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).