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Convergence over fractals for the periodic Schrödinger equation

Daniel Eceizabarrena and Renato Lucà

Vol. 15 (2022), No. 7, 1775–1805
Abstract

We consider a fractal refinement of Carleson’s problem for pointwise convergence of solutions to the periodic Schrödinger equation to their initial datum. For α (0,d] and

s < d 2(d + 1)(d + 1 α),

we find a function in Hs(𝕋d) whose corresponding solution diverges in the limit t 0 on a set with strictly positive α-Hausdorff measure. We conjecture this regularity threshold to be optimal. We also prove that

s > d 2(d + 2)(d + 2 α)

is sufficient for the solution corresponding to any datum in Hs(𝕋d) to converge to such datum α-almost everywhere.

Keywords
Schrödinger maximal function, pointwise convergence, Carleson problem
Mathematical Subject Classification
Primary: 35J10
Secondary: 42B37, 11J83, 11L07
Milestones
Received: 21 May 2020
Revised: 8 December 2020
Accepted: 19 March 2021
Published: 5 December 2022
Authors
Daniel Eceizabarrena
Department of Mathematics and Statistics
University of Massachusetts
Amherst, MA
United States
Renato Lucà
Basque Center for Applied Mathematics
Bilbao
Spain
Ikerbasque
Basque Foundation for Science
Bilbao
Spain