The Capacitary John-Nirenberg Inequality Revisited

TL;DR

This paper extends the classical John-Nirenberg inequality to translation invariant Hausdorff contents, establishing key estimates and structural conditions that characterize when capacities are equivalent to Hausdorff contents.

Contribution

It introduces a packing condition that characterizes when capacities are equivalent to Hausdorff contents and extends inequalities to general outer capacities satisfying this condition.

Findings

Packing condition characterizes capacity-Hausdorff content equivalence

Maximal function estimates established for Hausdorff contents

Extension of John-Nirenberg inequalities to outer capacities

Abstract

In this paper, we establish maximal function estimates, Lebesgue differentiation theory, Calder\'on-Zygmund decompositions, and John-Nirenberg inequalities for translation invariant Hausdorff contents. We further identify a key structural component of these results -- a packing condition satisfied by these Hausdorff contents which compensates for the non-linearity of the capacitary integrals. We prove that for any outer capacity, this packing condition is satisfied if and only if the capacity is equivalent to its induced Hausdorff content. Finally, we use this equivalence to extend the preceding theory to general outer capacities which are assumed to satisfy this packing condition.