Lebesgue points and capacities via boxing inequality in metric spaces

Abstract

Abstract. The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincaré inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.

Acharacterization of Newtonian functions with zero boundary values

“…This is shown for p > 1 by Theorem 4.1 in [22] and for p = 1 by Theorem 4.1 in [21]. By assumption (1.2), u * (x) = 0 for every x ∈ X \ .…”

“…Then u ∈ N 1, p (X ) and u = 0 in X \ . By Theorem 4.1 in [22] and Theorem 4.1 in [21], p-quasievery point of X is a Lebesgue point of u. Hence for p-quasievery x ∈ X \ , we have (B(x, r )) Remark 5.4 The assumption that is open can be replaced with the condition that is Borel measurable in Theorem 1.1.…”

The equivalence between pointwise Hardy inequalities and uniform fatness

“…Hence some information is inevitably lost once we pass from the pointwise p-Hardy inequality or uniform p-fatness (for 1 < p < ∞) to Hausdorff contents; in the case p = 1 there is indeed an equivalence, cf. [25]. However, by the self-improvement of the assertions of Theorem 2, we can still have the following equivalent characterization in terms of Hausdorff contents (see also [10,28]).…”

“…Estimate (12) follows easily from a standard telescoping argument, see for example [13]. Note that u has Lebesgue points almost everywhere in the p-capacity sense, see [23,25]. …”

“…[18,19]. In metric spaces we follow [1,2,25], and use Hausdorff contents H t R , defined by applying the Carathéodory construction to functions…”