On the endpoint estimate for discrete spherical average over sparse sequences
Let $d\geq5$. For a strictly increasing sequence $(μ_k)$ of positive integers, set $λ_k=μ_k!$ and consider the lacunary discrete spherical maximal operator $A_\star f:=\sup_k |A_{λ_k}f|$ associated with the discrete spherical averages \[ A_λf(x):=\frac1{s_λ}\sum_{\substack{n\in\mathbb{Z}^d,\\ |n|^2=λ}} f(x-n), \] where $s_λ:=\#\{n\in\mathbb{Z}^d:|n|^2=λ\}$. Kesler, Lacey and Mena proved that $A_\star$ is bounded on $\ell^p(\mathbb{Z}^d)$ for every $p>1$ if $\logμ_k/\log k\longrightarrow\infty$, and asked about its endpoint behavior at $\ell\log\ell$. We resolve this endpoint question by characterizing all factorial sequences for which the $\ell\log\ell$ estimate holds. Define \[ C_{\log}=\sup_{N\geq2}\frac{\#\left\{k\geq 1:μ_k\leq N\right\}}{1+\log N}. \] We prove that the $\ell\log\ell$ endpoint estimate holds if and only if $C_{\log}<\infty$. More precisely, if $C_{\log}<\infty$, then for every $α>0$ and every finitely supported $f:\mathbb{Z}^d\to\mathbb C$, \begin{align*} \#\{x\in\mathbb{Z}^d:A_\star f(x)>α\} \leq C_d(1+C_{\log})\sum_x \frac{|f(x)|}α \left(1+\log^+\frac{|f(x)|}α\right), \end{align*} where $C_d$ depends only on $d$. Conversely, if the above inequality holds with a finite constant $C_0$ in place of $C_d(1+C_{\log})$, then $C_{\log}\leq C_d(1+C_0)$.