Lacunary Discrete Spherical Maximal Functions

TL;DR

This paper establishes new bounds for discrete spherical averages over lacunary radii in high dimensions, improving understanding of maximal functions and their boundedness properties in discrete harmonic analysis.

Contribution

It provides novel $ ext{ell}^p$ bounds for lacunary discrete spherical maximal functions in dimensions $d geq 5$, extending previous results to more general radii sets.

Findings

Boundedness of the maximal operator for $p$ in the specified range

Applicable to lacunary sets of radii with $ ext{ell}^p$ norms

Decomposition approach simplifies endpoint estimates

Abstract

We prove new $\ell ^{p} (\mathbb Z ^{d})$ bounds for discrete spherical averages in dimensions $ d \geq 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 _{\lambda } f $ is the spherical average of $ f$ over the discrete sphere of radius $ \lambda $, we have \begin{equation*} \bigl\lVert \sup _{k} \lvert A _{\lambda _k} f \rvert \bigr\rVert _{\ell ^{p} (\mathbb Z ^{d})} \lesssim \lVert f\rVert _{\ell ^{p} (\mathbb Z ^{d})}, \qquad \tfrac{d-2} {d-3} < p \leq \tfrac{d} {d-2},\ d\geq 5, \end{equation*} for any lacunary sets of integers $ \{\lambda _k ^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.