The higher-dimensional Sobolev space-time estimate for the Schrödinger operator

Let $U$ denote the Schrödinger operator, let $B^p_{\beta,p}(\mathbb R^d)$ be the Besov space with smoothness $\beta$, and let $I$ be the time interval. For $p\in[2,\infty)$ and $r\in[2,\infty]$, assume

$$\frac{d}{p}+\frac{1}{r}<\frac{d}{2}$$

and

$$\frac{2d+1}{p}+\frac{1}{r}<d.$$

Higher-dimensional space-time estimate. Then

$$U:B^p_{\alpha,p}(\mathbb R^d)\to L^p(\mathbb R^d;L^r(I))$$

is bounded, where

$$\alpha=d\left(1-\frac{2}{p}\right)-\frac{2}{r}.$$

This is proposed as the higher-dimensional analogue of the sharp one- and two-dimensional estimates established earlier in the paper. The parser supplies no evidence that the estimate has been proved or disproved, so its status remains open.