正交系的临界Strichartz估计
Critical Strichartz estimates for orthonormal systems
AI总结：
本文针对齐次索伯列夫空间中正交系的Strichartz估计公开问题，证明$q=4$且$r<\infty$的临界情形下$\alpha=q/2$时估计成立，方法可推广至分数阶薛定谔传播子。
AI中文摘要：
自由薛定谔传播子的正交Strichartz估计形式为：对齐次索伯列夫空间$\text{\dot{H}}^s(\mathbb{R}^d)$中的任意正交系$(f_j)_j$，有$\left\Vert\sum_j\lambda_j\left|e^{it\Delta}f_j\right|^2\right\Vert_{L_t^\frac{q}{2}L_x^\frac{r}{2}(\mathbb{R}\times\mathbb{R}^d)}\lesssim\Vert\lambda\Vert_{\ell^\alpha(\mathbb{C})}$。在可容许区域内，当$q\geq r/d'$时已确定$\alpha$的最优范围；当$d\geq2$且$2<q<r/d'$时，$\alpha=q/2$时该估计是否成立是公开问题。本文证明，当$q=4$且$r<\infty$时，该临界情形下估计成立，且方法具有普适性，可推广到分数阶薛定谔传播子。
英文摘要：
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 $α$ has been established when $q\geq r/d'$. For $d\geq2$ and $2<q<r/d'$, it has been an open question as to determine whether the estimate holds when $α=q/2$. We prove that the estimate holds in this critical case whenever $q=4$ and $r<\infty$. Our approach is robust and we illustrate this by extending the result to fractional Schrödinger propagators.
