Communications on Pure and Applied Analysis (CPAA)

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

Pages: 881 - 898, Volume 12, Issue 2, March 2013      doi:10.3934/cpaa.2013.12.881

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Roberta Ghezzi - CMAP, École Polytechnique -- Team GECO, INRIA Saclay, France (email)
Frédéric Jean - ENSTA ParisTech, UMA, and Team GECO, INRIA Saclay -- Île-de-France, Paris, France (email)

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 $(H^k,1)$-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.

Keywords:  Rectifiability, Hausdorff measures, non Euclidean metric spaces, Hölder maps, sub-Riemannian geometry.
Mathematics Subject Classification:  Primary: 28A78, 30L99 ; Secondary: 53C17.

Received: September 2011;      Revised: June 2012;      Available Online: September 2012.