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Coarea integration in metric spaces

Malý, Jan (2003)

Nonlinear Analysis, Function Spaces and Applications

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 ^ m ( E u - 1 ( y ) ) = 0 for m -a.e. y Y , where m is the m -dimensional Hausdorff measure and ^ 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...

Comparison of Hausdorff measures with respect to the Euclidean and the Heisenberg metric.

Zoltán M. Balogh, Matthieu Rickly, Francesco Serra Cassano (2003)

Publicacions Matemàtiques

We compare the Hausdorff measures and dimensions with respect to the Euclidean and Heisenberg metrics on the first Heisenberg group. The result is a dimension jump described by two inequalities. The sharpness of our estimates is shown by examples. Moreover a comparison between Euclidean and H-rectifiability is given.

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