Critical Strichartz estimates for orthonormal systems

Orthonormal Strichartz estimates for the free Schrödinger propagator take the form \begin{equation*} \left\Vert\sum_jλ_j\left|e^{itΔ}f_j\right|^2\right\Vert_{L_t^\frac{q}{2}L_x^\frac{r}{2}($\mathbb{R}$\times$\mathbb{R}$^d)}\lesssim\Vertλ\Vert_{\ell^α($\mathbb{C}$)} \end{equation*}for arbitrary orthonormal systems $(f_j{)}_j$ in the homogeneous Sobolev space ${\dot{H}}^s({\mathbb{R}}^d)$. In the admissible region, the optimal range of $\alpha$ has been established when $q\ge r\mathrm{/}{d}^{\mathrm{\prime }}$. For $d\ge 2$ and $2<q<r\mathrm{/}{d}^{\mathrm{\prime }}$, it has been an open question as to determine whether the estimate holds when $\alpha =q\mathrm{/}2$. We prove that the estimate holds in this critical case whenever $q=4$ and $r<\mathrm{\infty }$. Our approach is robust and we illustrate this by extending the result to fractional Schrödinger propagators.