Maximal estimates for orthonormal systems of wave equations with sharp regularity

Abstract

We study maximal estimates for the wave equation with orthonormal initial data. In dimension $d=3$, we establish optimal results with the sharp regularity exponent up to the endpoint. In higher dimensions $d\ge 4$ and also in $d=2$, we obtain sharp bounds for the Schatten exponent (summability index) ${\hat{I}}^2\in [2,\infty ]$ when $d\ge 4$, and ${\hat{I}}^2\in [1,2]$ when $d=2$, improving upon the previous estimates due to Kinoshita--Ko--Shiraki. Our approach is based on a novel analysis of a key integral arising in the case ${\hat{I}}^2=2$, which allows us to refine existing techniques and achieve the optimal estimates.

16 pages, 5 figures