Approximations of Sobolev norms in Carnot groups

Details Approximations of Sobolev norms in Carnot groups Approximations of Sobolev norms in Carnot groups

2010-06-29

arxiv.org/abs/1006.5522v1

Mathematics

Functional Analysis

Scientific paper

10.1142/S0219199711004439

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a Poincar\'e-type inequality. The main results that we present are a generalization of these two works to a non-Euclidean setting, namely that of Carnot groups. We show that the seminorm expressd in terms of the intrinsic distance is equivalent to the $L^p$ norm of the intrinsic gradient, and provide a Poincar\'e-type inequality on Carnot groups by means of a constructive approach which relies on one-dimensional estimates. Self-improving properties are also studied for some cases of interest.

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