2018/10/29 by Kesler, Robert, Lacey, Michael T., Mena, Dario
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.1810.12344
We prove new ℓ p (\mathbb Z d) bounds for discrete spherical averages in dimensions d ≥ 5. We focus on the case of lacunary radii, first for general lacunary radii, and then for certain kinds of highly composite choices of radii. In particular, if A λ f is the spherical average of f over the discrete sphere of radius λ, we have ‖ sup k | A λk f | ‖ ℓ p (\mathbb Z d) \lesssim ‖ f‖ ℓ p (\mathbb Z d), \tfracd-2 d-3 lt; p ≤ \tfracd d-2, d≥ 5, for any lacunary sets of integers \λk 2 \. We follow a style of argument from our prior paper, addressing the full supremum. The relevant maximal operator is decomposed into several parts; each part requires only one endpoint estimate.