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Journal Articles Communications on Pure and Applied Analysis Year : 2012

A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces

Frédéric Jean

Abstract

The main motivation of this paper arises from the study of Carnot--Carathéodory spaces, where the class of $1$-rectifiable sets does not contain smooth non-horizontal curves; therefore a new definition of rectifiable sets including non-horizontal curves is needed. This is why we introduce in any metric space a new class of curves, called continuously metric differentiable of degree $k$, which are Hölder but not Lipschitz continuous when $k>1$. Replacing Lipschitz curves by this kind of curves we define $({\cal H}^k,1)$-rectifiable sets and show a density result generalizing the corresponding one in Euclidean geometry. This theorem is a consequence of computations of Hausdorff measures along curves, for which we give an integral formula. In particular, we show that both spherical and usual Hausdorff measures along curves coincide with a class of dimensioned lengths and are related to an interpolation complexity, for which estimates have already been obtained in Carnot--Carathéodory spaces.
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Dates and versions

hal-00623647 , version 1 (14-09-2011)
hal-00623647 , version 2 (24-05-2012)

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Roberta Ghezzi, Frédéric Jean. A new class of $({\cal H}^k,1)$-rectifiable subsets of metric spaces. Communications on Pure and Applied Analysis, 2012, 12 (2), pp.881-898. ⟨10.3934/cpaa.2013.12.881⟩. ⟨hal-00623647v2⟩
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