Lebesgue points and capacities via the boxing inequality in metric spaces
Lebesgue points and capacities via the boxing inequality in metric spaces
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DOI：
10.1512/iumj.2008.57.3168
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发表时间：
2008
影响因子：
1.1
通讯作者：
J. Kinnunen;R. Korte;N. Shanmugalingam;Heli Tuominen
中科院分区：
文献类型：
--
作者：
J. Kinnunen;R. Korte;N. Shanmugalingam;Heli Tuominen
本文研究了度量测度空间上Sobolev函数的正则性，该度量测度空间具有一个加倍测度，且支持一个弱Poincare不等式。证明了梯度可积为幂1的Sobolev函数在一个1-容量为零的集合外有Lebesgue点。我们还证明了1-容量等价于余维1的Hausdorff内容，并利用Frostman引理和有界变差函数研究了1-容量的特征。作为主要的技术工具，我们证明了一个度量空间版本的Gustin装箱不等式。我们的证明是基于覆盖参数和有界变差函数。周长测度、等周不等式和余面积公式在我们的方法中起着至关重要的作用。
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 Poincare 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.