On the endpoint estimate for discrete spherical average over sparse sequences

Abstract

Let $d\ge 5$, 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 \[ \#\{x:A_\star f(x)>α\} \le K\sum_x\frac{|f(x)|}α \left(1+\log^+\frac{|f(x)|}α\right)^γ\] holds with some finite $K$ for all $\alpha >0$ and finitely supported $f$ if and only if \[ \sup_{N\ge2}\frac{\#\{k:μ_k\le N\}}{(1+\log N)^γ}<\infty. \] 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 $1<p_{\ast }\le d/(d-1)$, we construct a set of squared radii whose maximal operator is of restricted weak type $(p_{\ast },p_{\ast })$, of strong type $(p,p)$ for $p>p_{\ast }$, and unbounded on ${\ell }^p$ for $1\le p<p_{\ast }$.

arXiv commentTypos are fixed and comments are welcome!