Sharp lower bounds for the periodic maximal Schr\"odinger operator in higher dimensions

Abstract

We prove sharp lower bounds for the Schr\"odinger maximal function on the torus T^d for all dimensions d at least 2, and as a corollary obtain sharp regularity conditions for pointwise convergence of the periodic Schr\"odinger equation. Combined with sufficiency results established by Compaan-Luc\'a-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schr\"odinger equation in all dimensions at least 2, and disproves the conjectured regularity condition of Miao-Yuan-Zhao, in contrast to the corresponding question in R^d. We also prove sharp estimates on the dimensions of divergence sets for the equation for high dimensional tori. Our approach uses complex multiplication on abelian varieties.