Strichartz estimates involving orthonormal systems at the critical summability exponent

TL;DR

This paper establishes new optimal orthonormal Strichartz estimates for the Schrödinger equation at the critical summability exponent, extending previous results and employing advanced interpolation techniques.

Contribution

It proves global strong-type orthonormal Strichartz estimates at the critical exponent, improving upon prior work and using novel restricted weak-type estimates and interpolation methods.

Findings

Established optimal orthonormal Strichartz estimates at critical exponent

Extended the range of known estimates for Schrödinger operators

Utilized real interpolation and restricted weak-type estimates effectively

Abstract

The primary objective of this paper is to investigate the orthonormal Strichartz estimates at the critical summability exponent for the Schr\"odinger operator $e^{it\Delta}$ with initial data from the homogeneous Sobolev space $\dot{H}^s (\mathbb{R}^n)$. We prove new global strong-type orthonormal Strichartz estimates in the interior of $ODCA$ at the optimal summability exponent $\alpha=q$, thereby substantially supplymenting the work of Bez-Hong-Lee-Nakamura-Sawano \cite{Bez-Hong-Lee-Nakamura-Sawano}. Our approach is based on restricted weak-type orthonormal estimates, real interpolation argument and the advantageous condition $q<p$ in the interior of $ODCA$.