Bounds for Lacunary maximal functions given by Birch–Magyar averages

Cook, Brian; Hughes, Kevin

Abstract

We obtain positive and negative results concerning lacunary discrete maximal operators defined by dilations of sufficiently nonsingular hypersurfaces arising from Diophantine equations in many variables. Our negative results show that this problem differs substantially from that of lacunary discrete maximal operators defined along a nonsingular hypersurface. Our positive results are improvements over bounds for the corresponding full maximal functions which were initially studied by Magyar.

In order to obtain positive results, we use an interpolation technique of the second author to reduce problem to a maximal function of main terms. The main terms take the shape of those introduced in work of the first author, which is a more localized version of the main terms that appear in work of Magyar. The main ingredient of this paper is a new bound on the main terms near []. For our negative results we generalize an argument of Zienkiewicz.

Citation

Cook, B., & Hughes, K. (2021). Bounds for Lacunary maximal functions given by Birch–Magyar averages. Transactions of the American Mathematical Society, 374(6), 3859-3879. https://doi.org/10.1090/tran/8152

• Journal Article Type: Article
• Acceptance Date: Feb 12, 2020
• Online Publication Date: Mar 26, 2021
• Publication Date: 2021
• Deposit Date: Nov 21, 2022
• Journal: Transactions of the American Mathematical Society
• Print ISSN: 0002-9947
• Electronic ISSN: 1088-6850
• Publisher: American Mathematical Society
• Peer Reviewed: Peer Reviewed
• Volume: 374
• Issue: 6
• Pages: 3859-3879
• DOI: https://doi.org/10.1090/tran/8152
• Keywords: Applied Mathematics; General Mathematics
• Public URL: http://researchrepository.napier.ac.uk/Output/2963232