Lp-improving for discrete spherical averages

Hughes, Kevin

Abstract

We initiate the theory of -improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove -improving estimates for the discrete spherical averages and some of their generalizations. As an application of our -improving inequalities for the dyadic discrete spherical maximal function, we give a new estimate for the full discrete spherical maximal function in four dimensions. Our proofs are analogous to Littman’s result on Euclidean spherical averages. One key aspect of our proof is a Littlewood–Paley decomposition in both the arithmetic and analytic aspects. In the arithmetic aspect this is a major arc-minor arc decomposition of the circle method.

Citation

Hughes, K. (2020). Lp-improving for discrete spherical averages. Annales Henri Lebesgue, 3, 959-980. https://doi.org/10.5802/ahl.50

• Journal Article Type: Article
• Acceptance Date: Feb 11, 2020
• Online Publication Date: Aug 24, 2020
• Publication Date: 2020
• Deposit Date: Nov 21, 2022
• Publicly Available Date: Nov 22, 2022
• Journal: Annales Henri Lebesgue
• Peer Reviewed: Peer Reviewed
• Volume: 3
• Pages: 959-980
• DOI: https://doi.org/10.5802/ahl.50
• Keywords: Lp-improving, discrete averages, discrete maximal functions, circle method, Littlewood–Paley theory
• Public URL: http://researchrepository.napier.ac.uk/Output/2963191
• Publisher URL: https://annales.lebesgue.fr/index.php/AHL/