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

Kevin Hughes

Published 2018-04-24Version 1

In this paper, we prove $\ell^p$-improving estimates for the discrete spherical averages and some of their generalizations. At first glance this problem appears trivial, but upon further examination we obtain interesting, nontrivial bounds. As an application we give a new estimate for the discrete spherical maximal function in four dimensions. We conclude by introducing a principle to describe the analogy with Littman's result for continuous spherical averages.