${\ell }^p$-improving for discrete spherical averages

Abstract

We initiate the theory of ${\ell }^p$-improving inequalities for arithmetic averages over hypersurfaces and their maximal functions. In particular, we prove ${\ell }^p$-improving estimates for the discrete spherical averages and some of their generalizations. As an application of our ${\ell }^p$-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.

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