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

We exhibit a range of ${\mathrm{\ell }}^p({\mathbb{Z}}^d)$-improving properties for the discrete spherical maximal average in every dimension $d\ge 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from ${\mathrm{\ell }}^2(w)$ into ${\mathrm{\ell }}^2(w)$ provided $w^{\frac{d}{d-4}+\delta }$ belongs to the Muckenhoupt class $A_2$ for some $\delta >0.$