Sharp ${\ell }^p$-Improving Estimates for Fixed-Radius Discrete Spherical Averages

Rui Han, Fan Yang

Let $d\ge 4$ and let $R>0$. When $d=4$, assume that $R^2\in \mathbb{N}\setminus 4\mathbb{N}$; when $d\ge 5$, let $R^2\in \mathbb{N}$ be arbitrary. We prove the fixed-radius estimat…