Lee–Rogers–Seeger maximal Schrödinger conjecture

Let $I$ be a compact interval. For $p,q\in[1,\infty]$, write $L^q_xL^\infty_t(\mathbb{R}\times I)$ for the corresponding mixed-norm space. For $\alpha\in\mathbb{R}$, let $B^q_{\alpha,q}(\mathbb{R})$ be the Besov space with norm

$$\|f\|_{B^q_{\alpha,q}(\mathbb{R})}=\left(\sum_{k\geq0}2^{k\alpha q}\|P_kf\|_{L^q(\mathbb{R})}^q\right)^{1/q},$$

where $P_k$ is the Fourier projection to $\{|\xi|\sim2^k\}$. Lee–Rogers–Seeger maximal Schrödinger conjecture. For $q>3$ and $\alpha=1-2/q$,

$$\|e^{it\Delta}f\|_{L^q_xL^\infty_t(\mathbb{R}\times I)}\leq C_{p,\alpha}\|f\|_{B^q_{\alpha,q}(\mathbb{R})}.$$

This conjecture concerns the $L^q$ boundedness of the maximal Schrödinger operator in the plane and implies regularity results related to pointwise convergence. The paper proves it when $q>18/5$, while the full range $q>3$ remains open.