Hardy's inequality for functions vanishing on a part of the boundary

We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.Comment: 26 pages, 2 figures, includes several improvements in Sections 6-8 allowing to relax the assumptions in the main results. Final version published at http://link.springer.com/article/10.1007%2Fs11118-015-9463-

Definitions of quasiconformality

We establish that the infinitesimal “ H -definition” for quasiconformal mappings on Carnot groups implies global quasisymmetry, and hence the absolute continuity on almost all lines. Our method is new even in R n where we obtain that the “limsup” condition in the H -definition can be replaced by a “liminf” condition. This leads to a new removability result for (quasi)conformal mappings in Euclidean spaces. An application to parametrizations of chord-arc surfaces is also given.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46582/1/222_2005_Article_BF01241122.pd