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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=
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