ℓ^p-improving inequalities for Discrete Spherical Averages

2018/04/26 by Kesler, Robert, Lacey, Michael T.
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1804.09845

Abstract

Let λ² ∈ \mathbb N , and in dimensions d≥ 5, let A_λ f (x) denote the average of f : \mathbb Z ^d → \mathbb R over the lattice points on the sphere of radius λ centered at x. We prove ℓ ^p improving properties of A_λ. ‖ A_λ‖_{ℓ ^p → ℓ ^p'} ≤ C_{d,p, ω(λ² )} λ^d ( 1-\frac2p), \tfracd-1d+1 lt; p ≤ \fracd d-2. 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 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.

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