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

Michael T. Lacey, Robert Kesler

Published 2018-04-26Version 1

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'}} \lesssim \lambda ^{-d (1 - \frac{2}p)}, \qquad \tfrac{d} {d-2} < p \leq 2. \end{equation*} The inequalities hold for the extended range $ \frac{d+1} {d-1} < p \leq \frac{d} {d-2}$ under the restriction that $ \lambda ^2 $ has a bounded number of distinct prime factors. We show that these inequalities cannot hold for $ p < \frac{d+1} {d-1}$. 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}$. The proof uses the decomposition of the corresponding multiplier whose properties were established by Magyar-Stein-Wainger, Magyar, and Hunt. Various endpoint estimates are combined with interpolation estimates of different types.