Orthonormal Strichartz inequalities and their applications on abstract measure spaces

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 $L^2(X,\hat{I}¼)$, where $(X,\hat{I}¼)$ is a measure space. Under the assumption that the kernel $K_{it}(x,y)$ of the Schrödinger 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 Schrödinger 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 $U(t)=e^{it\ddot{I}•(L)}\ddot{I}ˆ(\sqrt{L}),$ where $\ddot{I}•:{\mathbb{R}}^{+}\to \mathbb{R}$ is a smooth function and $\ddot{I}ˆ\in C_c^{\infty }([\frac{1}{2},2])$. 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 $S=\{(\hat{I}¾,\ddot{I}•(\hat{I}¾):\hat{I}¾\in \mathbb{R})\}$, where $\ddot{I}•$ satisfies certain growth condition.

40 pages