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# Besov Spaces and Hausdorff Dimension For Some Carnot-Carathéodory Metric Spaces

Published:2002-12-01
Printed: Dec 2002
• Leszek Skrzypczak
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## Abstract

We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero.
 Keywords: Besov spaces, sub-elliptic operators, Carnot-Carathéodory metric, Hausdorff dimension
 MSC Classifications: 46E35 - Sobolev spaces and other spaces of smooth'' functions, embedding theorems, trace theorems 43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc. 28A78 - Hausdorff and packing measures

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