Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function, M_Cf(x):=supr>0⁡[Formula presented] ∫_B(x,r)|f|dC, for C= the Hausdorff content or a Riesz capacity. For these maximal functions, we prove a strong-type (p,p) bound for 1<p≤+∞ on the capacitary integration spaces L^p(C) and a weak-type (1,1) bound on the capacitary integration space L¹(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.