Convergence over fractals for the Schr\"odinger equation

Abstract

We consider a fractal refinement of the Carleson problem for the Schr\"odinger equation, that is to identify the minimal regularity needed by the solutions to converge pointwise to their initial data almost everywhere with respect to the $\alpha$-Hausdorff measure ($\alpha$-a.e.). We extend to the fractal setting ($\alpha < n$) a recent counterexample of Bourgain \cite{Bourgain2016}, which is sharp in the Lebesque measure setting ($\alpha = n$). In doing so we recover the necessary condition from \cite{zbMATH07036806} for pointwise convergence~$\alpha$-a.e. and we extend it to the range $n/2<\alpha \leq (3n+1)/4$.