ℓp-Improving Inequalities for Discrete Spherical Averages

Abstract

Let λ2 ∈ ℕ, and in dimensions d ≥ 5, let Aλf(x) denote the average of f: ℤd → ℝ over the lattice points on the sphere of radius λ centered at x. We prove lp improving properties of Aλ: $$\begin{array}{*{20}{c}} {{{\left\| {{A_\lambda }} \right\|}_{{\ell ^{p \to }}{\ell ^{p'}}}} \leqslant {C_{d,p,\omega ({\lambda ^2})}}{\lambda ^{(1 - \tfrac{2}{p})}},}&{\frac{{d - 1}}{{d + 1}} < p \leqslant \frac{d}{{d - 2}}} \end{array}.$$ It holds in dimension d = 4 for odd λ2. The dependence is in terms of ω(λ2), the number of distinct prime factors of λ2. These inequalities are discrete versions of a classical inequality of Littman and Strichartz on the Lp improving property of spherical averages on ℝd. 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.