The endpoint case of Carleson's pointwise convergence problem for the Schrodinger equation: a.e. convergence for data in H^{n/(2(n+1))}

Planar theorem: for every $f\in H^{1/3}(\mathbb R^2)$ there is a full-measure set $E_f$ on which the Gaussian-regularised evolution has a limit for every $0<t<1$ and converges to $f$ as $t\downarrow0$. Higher-dimensional theorem: the same for $f\in H^{n/(2(n+1))}(\mathbb R^n)$, every $n\ge3$. With Bourgain's necessary condition this gives the exact range $s\ge n/(2(n+1))$ for $n\ge2$. Not shown: the endpoint local $L^3$ maximal inequality in the plane, sharp $L^p$ maximal estimates at the endpoint in higher dimensions, or results for other dispersive equations or convergence along curves.

Planar theorem: for every $f\in H^{1/3}(\mathbb R^2)$ there is a full-measure set $E_f$ on which the Gaussian-regularised evolution has a limit for every $0<t<1$ and converges to $f$ as $t\downarrow0$. Higher-dimensional theorem: the same for $f\in H^{n/(2(n+1))}(\mathbb R^n)$, every $n\ge3$. With Bourgain's necessary condition this gives the exact range $s\ge n/(2(n+1))$ for $n\ge2$. Not shown: the endpoint local $L^3$ maximal inequality in the plane, sharp $L^p$ maximal estimates at the endpoint in higher dimensions, or results for other dispersive equations or convergence along curves.