Coarea integration in metric spaces

Malý, Jan

  • Nonlinear Analysis, Function Spaces and Applications, Publisher: Czech Academy of Sciences, Mathematical Institute(Praha), page 149-192

Abstract

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Let X be a metric space with a doubling measure, Y be a boundedly compact metric space and u : X → Y be a Lebesgue precise mapping whose upper gradient g belongs to the Lorentz space L m , 1 , m ≥ 1 . Let E ⊂ X be a set of measure zero. Then ℋ ^ m ( E ∩ u - 1 ( y ) ) = 0 for ℋ m -a.e. y ∈ Y , where ℋ m is the m -dimensional Hausdorff measure and 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.

How to cite

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Malý, Jan. "Coarea integration in metric spaces." Nonlinear Analysis, Function Spaces and Applications. Praha: Czech Academy of Sciences, Mathematical Institute, 2003. 149-192. <http://eudml.org/doc/221662>.

@inProceedings{Malý2003,
abstract = {Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\rightarrow Y$ 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 $\widehat\{\mathcal \{H\}\}_m(E\cap u^\{-1\}(y))=0$ for $\mathcal \{H\}_m$-a.e. $y\in Y$, where $\mathcal \{H\}_m$ is the $m$-dimensional Hausdorff measure and $\widehat\{\mathcal \{H\}\}_m$ 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.},
author = {Malý, Jan},
booktitle = {Nonlinear Analysis, Function Spaces and Applications},
keywords = {coarea formula; Eilenberg inequality; Hausdorff content; Hausdorff measure; Lebesgue points; Riesz potentials; Lorentz space; upper gradient; Poincaré inequality; space of homogenous type; metric space; doubling measure},
location = {Praha},
pages = {149-192},
publisher = {Czech Academy of Sciences, Mathematical Institute},
title = {Coarea integration in metric spaces},
url = {http://eudml.org/doc/221662},
year = {2003},
}

TY - CLSWK
AU - Malý, Jan
TI - Coarea integration in metric spaces
T2 - Nonlinear Analysis, Function Spaces and Applications
PY - 2003
CY - Praha
PB - Czech Academy of Sciences, Mathematical Institute
SP - 149
EP - 192
AB - Let $X$ be a metric space with a doubling measure, $Y$ be a boundedly compact metric space and $u:X\rightarrow Y$ 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 $\widehat{\mathcal {H}}_m(E\cap u^{-1}(y))=0$ for $\mathcal {H}_m$-a.e. $y\in Y$, where $\mathcal {H}_m$ is the $m$-dimensional Hausdorff measure and $\widehat{\mathcal {H}}_m$ 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.
KW - coarea formula; Eilenberg inequality; Hausdorff content; Hausdorff measure; Lebesgue points; Riesz potentials; Lorentz space; upper gradient; Poincaré inequality; space of homogenous type; metric space; doubling measure
UR - http://eudml.org/doc/221662
ER -

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