Lacunary discrete spherical maximal functions

Abstract

We prove new l^p(Z^d) bounds for discrete spherical averages in dimensions d greater than or equal to 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 Aλf is the spherical average of f over the discrete sphere of radius λ, we have for any lacunary sets of integers {λ 2 k}. 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.