A NEW CLASS OF (H-k, 1)-RECTIFIABLE SUBSETS OF METRIC SPACES

作者：Ghezzi Roberta^*; Jean Frederic

来源：Communications on Pure and Applied Analysis, 2013, 12(2): 881-898.

DOI：10.3934/cpaa.2013.12.881

• Rectifiability
• Hausdorff measures
• non Euclidean metric spaces
• Holder maps
• sub-Riemannian geometry

The main motivation of this paper arises from the study of Carnot-Caratheodory 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 Holder 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-Caratheodory spaces.

• 出版日期2013-3
• 单位INRIA