Orthonormal Strichartz inequalities and their applications on abstract measure spaces

TL;DR

This paper extends Strichartz inequalities to orthonormal systems on measure spaces, establishing new estimates for Schrödinger operators and applications to PDEs like the Hartree equation.

Contribution

It introduces orthonormal Strichartz estimates for a broad class of operators and applies these to well-posedness and restriction theorems, extending prior single-function results.

Findings

Established orthonormal Strichartz estimates for Schrödinger propagators.

Proved well-posedness of the Hartree equation in Schatten spaces.

Extended restriction theorems for Fourier transforms on specific hypersurfaces.

Abstract

The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,\mu)$, where $(X,\mu)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schr\"{o}dinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form \begin{equation*} \sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, \text{ for some }n\geq1, \end{equation*} where $T_0\in(0,+\infty]$, we establish Strichartz estimates for the Schr\"{o}dinger propagator $e^{itL}$ and using a duality principle argument by Frank-Sabin \cite{FS}, we extend it for a system of infinitely many fermions on $L^2(X)$. We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup $U(t)=e^{it\phi(L)}\psi(\sqrt{L}),$ where $\phi: \mathbb{R}^+\rightarrow \mathbb{R}$ is a smooth function and $\psi\in C_c^\infty([\frac{1}{2},2])$. As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces. In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega \cite{Kenig-Ponce-Vega} for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form $S=\{(\xi, \phi(\xi): \xi\in \mathbb{R})\}$, where $\phi$ satisfies certain growth condition.