Coarea integration in metric spaces

Malý, Jan

Nonlinear Analysis, Function Spaces and Applications, GDML_Books, (2003), p. 149-192 / Harvested from

Résumé

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 $L_{m, 1}$ , $m \geq 1$ . Let $E \subset X$ be a set of measure zero. Then ${\hat{ℋ}}_{m} (E \cap u^{- 1} (y)) = 0$ for $ℋ_{m}$ -a.e. $y \in Y$ , where $ℋ_{m}$ is the $m$ -dimensional Hausdorff measure and ${\hat{ℋ}}_{m}$ 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.

EUDML-ID : urn:eudml:doc:221662
Mots clés:
Mots clés:

@article{702476,
     title = {Coarea integration in metric spaces},
     booktitle = {Nonlinear Analysis, Function Spaces and Applications},
     series = {GDML\_Books},
     publisher = {Czech Academy of Sciences, Mathematical Institute},
     address = {Praha},
     year = {2003},
     pages = {149-192},
     url = {http://dml.mathdoc.fr/item/702476}
}

Malý, Jan. Coarea integration in metric spaces, dans Nonlinear Analysis, Function Spaces and Applications, GDML_Books,  (2003), pp. 149-192. http://gdmltest.u-ga.fr/item/702476/

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