A sharp covering theorem and Solyanik estimates for Euclidean balls

For every finite family of Euclidean balls in ${\mathbb{R}}^n$, $n\ge 2$, and every $0<\delta <1\mathrm{/}2$, we select a subfamily whose $(1+\delta )$-dilations cover the original union and whose undilated balls have multiplicity at most $C_n{\delta }^{-(n-1)\mathrm{/}2}$. This proves the covering estimate conjectured by Han and Lu \cite{HL}. As an application, we determine the optimal Solyanik asymptotic $${\mathcal{C}}_n(\alpha )-1{\asymp }_n(1-\alpha {)}^{2\mathrm{/}(n+1)}(\alpha \uparrow 1)$$ for the uncentered Hardy–Littlewood maximal operator over Euclidean balls, which is Conjecture~1(b) of Hagelstein and Parissis \cite{HP14}. For the modified uncentered maximal operators, we determine the optimal weak $(1,1)$ growth rate $(k-1{)}^{-(n-1)\mathrm{/}2}$ as $k\downarrow 1$, uniformly over Radon measures. We also obtain the corresponding $L^p$ and Fefferman–Stein bounds.