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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
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
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.
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