On the endpoint estimate for discrete spherical average over sparse sequences

Abstract

Let $d\ge5$, and let $A_\lambda$ denote the normalized discrete spherical average on $\mathbb{Z}^d$ with squared radius $\lambda$. For a strictly increasing sequence $(\mu_k)$ of positive integers, set $\lambda_k=\mu_k!$ and consider the lacunary maximal operator $A_\star f:=\sup_k|A_{\lambda_k}f|$. For each fixed $\gamma>0$, we prove that the estimate \[ \#\{x:A_\star f(x)>\alpha\} \le K\sum_x\frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right)^\gamma \] holds with some finite $K$ for all $\alpha>0$ and finitely supported $f$ if and only if \[ \sup_{N\ge2}\frac{\#\{k:\mu_k\le N\}}{(1+\log N)^\gamma}<\infty. \] The summand is interpreted as zero when $f(x)=0$. The case $\gamma=1$ answers the $\ell\log\ell$ endpoint question of Kesler, Lacey and Mena in the formulation stated above. For $\mu_k=2^{2^k}$, these estimates hold for every fixed $\gamma>0$, but $A_\star$ is not of weak type $(1,1)$. For factorial blocks formed from arbitrary subsets of dyadic grids at widely separated scales, we characterize restricted weak type $(p,p)$, $1<p\le d/(d-1)$, by a uniform covering condition. In particular, for each $1<p_*\le d/(d-1)$, we construct a set of squared radii whose maximal operator is of restricted weak type $(p_*,p_*)$, of strong type $(p,p)$ for $p>p_*$, and unbounded on $\ell^p$ for $1\le p<p_*$.