${\ell }^p({\mathbb{Z}}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Means, Revisited

Abstract

We prove an expanded range of ${\ell }^p({\mathbb{Z}}^d)$-improving properties and sparse bounds for discrete spherical maximal means in every dimension $d\ge 6$. Essential elements of the proofs are bounds for high exponent averages of Ramanujan and restricted Kloosterman sums.