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Potential Analysis

, Volume 45, Issue 2, pp 201–227 | Cite as

Bessel Potentials in Ahlfors Regular Metric Spaces

  • Miguel Andrés Marcos
Article

Abstract

In this paper we introduce Bessel potentials and the Sobolev potential spaces resulting from them in the context of Ahlfors regular metric spaces. The Bessel kernel is defined using a Coifman type approximation of the identity, and we show integration against it improves the regularity of Lipschitz, Besov and Sobolev-type functions. For potential spaces, we prove density of Lipschitz functions, and several embedding results, including Sobolev-type embedding theorems. Finally, using singular integrals techniques such as the T1 theorem, we find that for small orders of regularity Bessel potentials are inversible, its inverse in terms of the fractional derivative, and show a way to characterize potential spaces, concluding that a function belongs to the Sobolev potential space if and only if itself and its fractional derivative are in L p . Moreover, this characterization allows us to prove these spaces in fact coincide with the classical potential Sobolev spaces in the Euclidean case.

Keywords

Bessel potential Ahlfors spaces Fractional derivative Sobolev spaces Besov spaces 

Mathematics Subject Classification (2010)

43A85 

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Instituto de Matemática Aplicada del Litoral, UNL, CONICET, FIQSanta FeArgentina

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