Capacitary Maximal Inequalities and Applications

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 $1<p\le +\mathrm{\infty }$ 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.