  In this paper we introduce capacitary analogues of the Hardy-Littlewood
maximal function, \begin{align*} \mathcal{M}_C(f)(x):= \sup_{r>0}
\frac{1}{C(B(x,r))} \int_{B(x,r)} |f|\;dC, \end{align*} for $C=$ the Hausdorff
content or a Riesz capacity. For these maximal functions, we prove a
strong-type $(p,p)$ bound for $1<p \leq+\infty$ on the capacitary integration
spaces $L^p(C)$ and a weak-type $(1,1)$ bound on the capacitary integration
space $L^1(C)$. We show how these estimates clarify and improve the existing
literature concerning maximal function estimates on capacitary integration
spaces. As a consequence, we deduce correspondingly stronger differentiation
theorems of Lebesgue-type, which in turn, by classical capacitary inequalities,
yield more precise estimates concerning Lebesgue points for functions in
Sobolev spaces.
