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

Abstract

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 estimate $$\Vert A_{R}f{\Vert }_{{\ell }^{p^{\prime }}({\mathbb{Z}}^d)}\le C_{d,p,\epsilon }{R}^{-d(2/p-1)+\epsilon }\Vert f{\Vert }_{{\ell }^p({\mathbb{Z}}^d)}$$ for $(d+2)/d\le p\le 2$, where $p^{\prime }$ is the Hölder conjugate exponent of $p$ and $A_R$ is the probability average over the lattice sphere of radius $R$. This extends the fixed-radius estimates of Kesler--Lacey and Hughes to the sharp lower endpoint $p=(d+2)/d$.