On the endpoint estimate for discrete spherical average over sparse sequences

Let $d\ge 5$. For a lacunary sequence of radii $\{\lambda^{1/2}_k\}$ in the \emph{highly composite} regime, that is $\lambda_k=\mu_k !$ with $\log \mu_k/\log k\rightarrow\infty$ 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 $$A_\lambda f(x):=\frac1{s_\lambda}\sum_{\substack{n\in\Z^d\\|n|^2=\lambda}} f(x-n), \qquad s_\lambda:=\#\{n\in\Z^d:|n|^2=\lambda\}.$$ Kesler, Lacey and Mena proved that $A_\star$ is bounded on $\ell^p(\Z^d)$ 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 $$\mathcal N_\mu(N)=\#\{k:\mu_k\leq N\}\quad\text{and}\quad \mathcal C_\mu(\theta)=\sup_{N\geq2}\frac{\mathcal N_\mu(N)}{N^{\theta}}.$$ Then, for every $\alpha>0$ \begin{align*} \#\{x\in\mathbb Z^d:A_\star f(x)>\alpha\} \lesssim_d \sum_{x\in\mathbb Z^d}\frac{|f(x)|}{\alpha} \left(1+\log^+\frac{|f(x)|}{\alpha}\right)^2 \Theta_\mu\!\left(1+\log^+\frac{|f(x)|}{\alpha}\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.