${\ell }^p$-improving inequalities for Discrete Spherical Averages

Abstract

Let ${\lambda }^2\in \mathbb{N}$, and in dimensions $d\ge 5$, let $A_{\lambda }f(x)$ denote the average of $f:{\mathbb{Z}}^d\to \mathbb{R}$ over the lattice points on the sphere of radius $\lambda$ centered at $x$. We prove ${\ell }^p$ improving properties of $A_{\lambda }$. \begin{equation*} \lVert A_{λ}\rVert_{\ell ^{p} \to \ell ^{p'}} \leq C_{d,p, ω(λ^2 )} λ^{d ( 1-\frac{2}p)}, \qquad \tfrac{d-1}{d+1} < p \leq \frac{d} {d-2}. \end{equation*} It holds in dimension $d=4$ for odd ${\lambda }^2$. The dependence is in terms of $\omega ({\lambda }^2)$, the number of distinct prime factors of ${\lambda }^2$. These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the $L^p$ improving property of spherical averages on ${\mathbb{R}}^d$, in particular they are scale free, in a natural sense. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar-Stein-Wainger, and Magyar. We then use a proof strategy of Bourgain, which dominates each part of the decomposition by an endpoint estimate.

10 pages