New bounds for discrete lacunary spherical averages

Abstract

We show that the discrete lacunary spherical maximal function is bounded on $l^p({\mathbb{Z}}^d)$ for all $p>\frac{d+1}{d-1}$. Our range is new in dimension 4, where it appears that little was previously known for general lacunary radii. Our technique follows that of Kesler-Lacey-Mena, using the Kloosterman refinement to improve the estimates in several places, which leads to an overall improvement in dimension 4.

arXiv commentWhile the authors have an improved version, there is still an error in showing the bound (2.7). The results in this paper hold for averages (currently in the literature) so we will not pursue this at this time

Cited by in corpus (1)

• Sharp $l^p$-improving estimates for the discrete paraboloid