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Acta Mathematica Sinica

, 22:1227 | Cite as

Towards an L p Potential Theory for Sub–Markovian Semigroups: Kernels and Capacities

  • Niels JacobEmail author
  • René L. Schilling
ORIGINAL ARTICLES

Abstract

We study in fairly general measure spaces (X, μ) the (non–linear) potential theory of L p sub–Markovian semigroups which are given by kernels having a density with respect to the underlying measure. In terms of mapping properties of the operators we provide sufficient conditions for the existence (and regularity) of such densities. We give various (dual) representations for several associated capacities and, in the corresponding abstract Bessel potential spaces, we study the role of the truncation property. Examples are discussed in the case of ℝ n , where abstract Bessel potential spaces can be identified with concrete function spaces.

Keywords

nonlinear potential theory (r, p)–capacity Bessel potential space gamma–transform sub–Markovian semigroup 

MR (2000) Subject Classification

31C15 31C45 31C25 35S05 47B34 47B65 47D07 60J35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Wales SwanseaSwansea SA2 8PPUnited Kingdom
  2. 2.FB 12—MathematikPhilipps-Universität MarburgMarburgGermany

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