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Nonlinear potential theory for Sobolev spaces on Carnot groups

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Abstract

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.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk, Russia

    S. K. Vodop’yanov

  2. Novosibirsk State University, Novosibirsk, Russia

    N. A. Kudryavtseva

Authors
  1. S. K. Vodop’yanov
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  2. N. A. Kudryavtseva
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Corresponding author

Correspondence to S. K. Vodop’yanov.

Additional information

To Yuriĭ Grigor’evich Reshetnyak on his 80th birthday.

Original Russian Text Copyright © 2009 Vodop’yanov S. K. and Kudryavtseva N. A.

The authors were supported by the Russian Foundation for Basic Research (Grant 08-01-00531) and the State Maintenance Program for the Leading Scientific Schools (Grant NSh-5682.2008.1).

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Novosibirsk. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 5, pp. 1016–1036, September–October, 2009.

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Vodop’yanov, S.K., Kudryavtseva, N.A. Nonlinear potential theory for Sobolev spaces on Carnot groups. Sib Math J 50, 803–819 (2009). https://doi.org/10.1007/s11202-009-0091-7

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