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Lp-improving for discrete spherical averages

Hughes, Kevin


Kevin Hughes


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.

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
Keywords Lp-improving, discrete averages, discrete maximal functions, circle method, Littlewood–Paley theory
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