Capacitary maximal inequalities and applications

You Wei Benson Chen, Keng Hao Ooi, Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


In this paper we introduce capacitary analogues of the Hardy-Littlewood maximal function, MCf(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 Lp(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.

Original languageEnglish
Article number110396
JournalJournal of Functional Analysis
Issue number12
Publication statusPublished - 2024 Jun 15


  • Capacitary integration
  • Capacitary maximal functions
  • Maximal function inequalities
  • Strong-type capacitary inequality

ASJC Scopus subject areas

  • Analysis


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