Critical orthonormal Strichartz estimate for the Schrödinger propagator

Determine whether the orthonormal Strichartz estimate for the free Schrödinger propagator on \(\mathbb{R}^d\), with \(d\geq 2\), holds at the critical exponent \(\alpha=q/2\) for the admissible range \(2<q<r/d'\), beyond the case \(q=4\) and \(r<\infty\) established in the paper.

Background

The paper studies orthonormal extensions of homogeneous Strichartz estimates for the free Schrödinger evolution. Existing results determine the optimal sequence-space exponent in a substantial portion of the admissible region, but leave the critical exponent $\alpha=q/2$ unresolved in the region corresponding to $d\geq2$ and $2

The paper proves the critical estimate when $q=4$ and $r<\infty$, corresponding to the segment $(M,N)$ in the authors’ parameter diagram. The broader determination of whether the critical estimate holds throughout the remaining region is therefore left unresolved.