On the endpoint estimate for discrete spherical average over sparse sequences
Let $d\geq5$. For a strictly increasing sequence $(\mu_k)$ of positive integers, set $\lambda_k=\mu_k!$ and consider the lacunary discrete spherical maximal operator $A_\star f:=\sup_k |A_{\lambda_k}f|$ associated with the discrete spherical averages \[ A_\lambda f(x):=\frac1{s_\lambda}\sum_{\substack{n\in\mathbb Z^d,\ |n|^2=\lambda}} f(x-n), \] where $s_\lambda:=\#\{n\in\mathbb Z^d:|n|^2=\lambda\}$. Kesler, Lacey and Mena proved that $A_\star$ is bounded on $\ell^p(\mathbb Z^d)$ for every $p>1$ if $\log\mu_k/\log k\longrightarrow\infty$, 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 \[ C_{\log}=\sup_{N\geq2}\frac{\#\left\{k\geq 1:\mu_k\leq N\right\}}{1+\log N}. \] 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 $\alpha>0$ and every finitely supported $f:\mathbb Z^d\to\mathbb C$, \begin{align*} \#\{x\in\mathbb Z^d:A_\star f(x)>\alpha\} \leq C_d(1+C_{\log})\sum_x \frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\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 $C_{\log}\leq C_d(1+C_0)$. Thus their growth condition alone is insufficient at this endpoint. In particular, the estimate holds for $\lambda_k=(2^k)!$ and fails for $\lambda_k=(k+\lceil\exp(\sqrt{k})\rceil)!$.