On the endpoint estimate for discrete spherical average over sparse sequences

Abstract

Let $d\ge 5$. For a lacunary sequence of radii $\{{\lambda }_k^{1/2}\}$ in the \emph{highly composite} regime, that is ${\lambda }_k={\mu }_k!$ with $\log {\mu }_k/\log k\to \infty$ as $k\to \infty$, we consider the lacunary discrete spherical maximal operator $A_{\star }f:={\sup }_k|A_{{\lambda }_k}f|$ associated with the discrete spherical averages $$A_{\lambda }f(x):=\frac{1}{s_{\lambda }}\sum _{\begin{array}{c}n\in {\mathbb{Z}}^d\\ |n{|}^2=\lambda \end{array}}f(x-n),s_{\lambda }:=\#\{n\in {\mathbb{Z}}^d:|n{|}^2=\lambda \}.$$ Kesler, Lacey and Mena proved that $A_{\star }$ is bounded on ${\ell }^p({\mathbb{Z}}^d)$ for each $p>1$, and raised the question on the endpoint behavior on the scale of Orlicz spaces. In this paper, we address this questions via establishing the following sequence-adapted endpoint estimate. Define $${\mathcal{N}}_{\mu }(N)=\#\{k:{\mu }_k\le N\}\text{and}{\mathcal{C}}_{\mu }(\theta )=\underset{N\ge 2}{\sup }\frac{{\mathcal{N}}_{\mu }(N)}{N^{\theta }}.$$ Then, for every $\alpha >0$ \begin{align*} \#\{x\in\mathbb Z^d:A_\star f(x)>α\} \lesssim_d \sum_{x\in\mathbb Z^d}\frac{|f(x)|}α \left(1+\log^+\frac{|f(x)|}α\right)^2 Θ_μ\!\left(1+\log^+\frac{|f(x)|}α\right), \end{align*} where ${\Theta }_{\mu }(L)=1+{\mathcal{C}}_{\mu }(c_d/L{)}^{3/2}$ and $c_d$ is a dimensional constant. We also remove the quantity $\ThetaMu$ and give the $\ell (\log \ell {)}^2\to {\ell }^{1,\infty }$ estimate when ${\lambda }_k=(2^k)!$. To the best of our knowledge, this provides the first endpoint estimate of this Orlicz weak type for $A_{\star }f$ as raised by Kesler, Lacey and Mena.