Lacunary Discrete Spherical Maximal Functions

Abstract

We prove new ${\ell }^p({\mathbb{Z}}^d)$ bounds for discrete spherical averages in dimensions $d\ge 5$. 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 _{λ_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.

13 pages, 1 Figure. v3. Section added to illustrate the proof technique in the continuous case. Final version, to appear in NYJM

References in corpus (1)

• Improved endpoint bounds for the lacunary spherical maximal operator

Cited by in corpus (1)

• Averages along the Square Integers: ${\ell }^p$ improving and Sparse Inequalities