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

Let $\lambda ^2 \in \mathbb N$, and in dimensions $d\geq 5$, 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_{\lambda }\rVert_{\ell ^{p} \to \ell ^{p'}} \leq C_{d,p, \omega (\lambda ^2 )} \lambda ^{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.