Integral representations and the generalized Poincaré inequality on Carnot groups.
Plotnikova, E.A. (2008)
Sibirskij Matematicheskij Zhurnal
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Displaying similar documents to “Nonlinear potential theory for Sobolev spaces on Carnot groups.”
Integral representations and the generalized Poincaré inequality on Carnot groups.
A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries
Enrico Le Donne (2017)
Analysis and Geometry in Metric Spaces
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Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory...
Trace inequalities for Carnot-Carathéodory spaces and applications
Donatella Danielli, Nicola Garofalo, Duy-Minh Nhieu (1998)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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Weighted Rellich inequality on -type groups and nonisotropic Heisenberg groups.
Jin, Yongyang, Han, Yazhou (2010)
Journal of Inequalities and Applications [electronic only]
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How to get rid of one of the weights in a two-weight Poincaré inequality?
Bruno Franchi, Piotr Hajłasz (2000)
Annales Polonici Mathematici
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We prove that if a Poincaré inequality with two different weights holds on every ball, then a Poincaré inequality with the same weight on both sides holds as well.
Representation formulas and weighted Poincaré inequalities for Hörmander vector fields
Bruno Franchi, Guozhen Lu, Richard L. Wheeden (1995)
Annales de l'institut Fourier
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We derive weighted Poincaré inequalities for vector fields which satisfy the Hörmander condition, including new results in the unweighted case. We also derive a new integral representation formula for a function in terms of the vector fields applied to the function. As a corollary of the versions of Poincaré’s inequality, we obtain relative isoperimetric inequalities.
Rectifiability and perimeter in step 2 Groups
Bruno Franchi, Raul Serapioni, Francesco Serra Cassano (2002)
Mathematica Bohemica
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We study finite perimeter sets in step 2 Carnot groups. In this way we extend the classical De Giorgi’s theory, developed in Euclidean spaces by De Giorgi, as well as its generalization, considered by the authors, in Heisenberg groups. A structure theorem for sets of finite perimeter and consequently a divergence theorem are obtained. Full proofs of these results, comments and an exhaustive bibliography can be found in our preprint (2001).
Hölder quasicontinuity of Sobolev functions on metric spaces.
Piotr Hajlasz, Juha Kinnunen (1998)
Revista Matemática Iberoamericana
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We prove that every Sobolev function defined on a metric space coincides with a Hölder continuous function outside a set of small Hausdorff content or capacity. Moreover, the Hölder continuous function can be chosen so that it approximates the given function in the Sobolev norm. This is a generalization of a result of Malý [Ma1] to the Sobolev spaces on metric spaces [H1].