Capacitary maximal inequalities and applications

In this paper we introduce capacitary analogues of the HardyLittlewood maximal function, M-C f (x) := sup(r> 0) 1/C(B (x, r)) integral(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 <= +infinity 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. (c) 2024 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).