Lee–Rogers–Seeger local space-time conjecture

Let $n\geq1$, let $f$ be a function on $\mathbb{R}^n$, and let $B_{\alpha,p}^p(\mathbb{R}^n)$ be the non-homogeneous Besov space with norm

$$\|f\|_{B_{\alpha,p}^p}=\left(\sum_{k\geq0}2^{k\alpha p}\|P_kf\|_{p}^p\right)^{1/p}.$$

For $2\leq p<\infty$ and $2\leq r\leq\infty$ satisfying

$$\frac{n}{p}+\frac{1}{r}<\frac{n}{2},\qquad \frac{2n+1}{p}+\frac{1}{r}<n,$$

Lee–Rogers–Seeger local space-time conjecture. The estimate

$$\|e^{it\Delta}f\|_{L_x^pL_t^r(\mathbb{R}^n\times[0,1])}\lesssim\|f\|_{B_{\alpha,p}^p(\mathbb{R}^n)},\qquad \alpha=n\left(1-\frac{2}{p}\right)-\frac{2}{r},$$

holds. The conjecture was proved for $n=1$, while the higher-dimensional case remains open; partial results follow from square-function and Fourier-restriction estimates.