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Title: Coarea integration in metric spaces (English)
Author: Malý, Jan
Language: English
Journal: Nonlinear Analysis, Function Spaces and Applications
Volume: Vol. 7
Issue: 2002
Pages: 149-192
Category: math
Summary: Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\to Y$ be a Lebesgue precise mapping whose upper gradient $g$ belongs to the Lorentz space $L_{m,1}$, $m\ge 1$. Let $E\subset X$ be a set of measure zero. Then $\widehat{\Cal H}_m(E\cap u^{-1}(y))=0$ for $\Cal H_m$-a.e.\ $y\in Y$, where $\Cal H_m$ is the $m$-dimensional Hausdorff measure and $\widehat{\Cal H}_m$ is the $m$-codimensional Hausdorff measure. This property is closely related to the coarea formula and implies a version of the Eilenberg inequality. The result relies on estimates of Hausdorff content of level sets of mappings between metric spaces and analysis of their Lebesgue points. Adapted versions of the Frostman lemma and of the Muckenhoupt-Wheeden inequality appear as essential tools. (English)
Keyword: coarea formula
Keyword: Eilenberg inequality
Keyword: Hausdorff content
Keyword: Hausdorff measure
Keyword: Lebesgue points
Keyword: Riesz potentials
Keyword: Lorentz space
Keyword: upper gradient
Keyword: Poincaré inequality
Keyword: space of homogenous type
Keyword: metric space
Keyword: doubling measure
MSC: 28A75
MSC: 28A78
MSC: 31C15
MSC: 42B25
MSC: 43A85
MSC: 46E35
MSC: 54E99
Date available: 2009-10-08T09:50:19Z
Last updated: 2012-08-03
Stable URL:
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