PreprintCC BY 4.0arXiv
Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions
Abstract
We prove sharp lower bounds for the Schrödinger 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 Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger 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.
Related
- Inverse spectral results for the magnetic Schrödinger operator in low dimensions
- Counterexamples for generalizations of the non-elliptic Schrödinger maximal operator
- Improved Lp bounds for the helical maximal function in dimensions n ≥ 5
- Weighted Liouville Theorems and A Priori Bounds for Indefinite Superlinear Equations with Pucci's Maximal Operator
- Sharp Lp--Lq bounds for the strong circular maximal operator
BibTeX
@misc{fenves2026sharp,
title = {{Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions}},
author = {Inbo Gottlieb Fenves and Jiahao Tan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32090},
eprint = {2609.32090},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32090},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp lower bounds for the periodic maximal Schrödinger operator in higher dimensions
AU - Inbo Gottlieb Fenves
AU - Jiahao Tan
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32090
UR - https://arxiv.org/abs/2609.32090
AB - We prove sharp lower bounds for the Schrödinger 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 Schrödinger equation. Combined with sufficiency results established by Compaan-Lucá-Staffilani, this yields a full resolution of Carleson's problem up to endpoint for the periodic Schrödinger 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.
ER -