$\ell^p$-improving for discrete spherical averages
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.
Keywords
Cite
@article{arxiv.1804.09260,
title = {$\ell^p$-improving for discrete spherical averages},
author = {Kevin Hughes},
journal= {arXiv preprint arXiv:1804.09260},
year = {2019}
}
Comments
18 pages, Comments welcome. 5 Oct 2018 submitted draft with referees' comments. In this version I used a more robust interpolation which improved the bounds. I emphasized the bounds for dyadic maximal functions appearing in the 1st version and streamlined the exposition for publication. An appendix on the range of $\ell^p$-boundedness in Magyar's Birch maximal functions was added