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Nonlinear potential theory for Sobolev spaces on Carnot groups. (Russian, English) Zbl 1224.46070

Sib. Mat. Zh. 50, No. 5, 1016-1036 (2009); translation in Sib. Math. J. 50, No. 5, 803-819 (2009).
Summary: Considering Bessel kernels on a Carnot group, we establish the main facts of nonlinear potential theory: a Wolff-type inequality, capacity estimates, and a strong capacity inequality. Deriving corollaries, we give an inequality of Sobolev-Adams type and relations between the capacity and Hausdorff measure, as well as lower bounds on the Teichmüller capacity. These yield the continuity of monotone functions of a Sobolev class and some estimates applicable to studying the fine properties of functions.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
31C45 Other generalizations (nonlinear potential theory, etc.)
43A70 Analysis on specific locally compact and other abelian groups
43A80 Analysis on other specific Lie groups
22E30 Analysis on real and complex Lie groups
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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