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On potentials in generalized Hölder spaces over uniform domains in ℝn

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Abstract

We show that Riesz-type potential operators of order α over uniform domains Ω in ℝn map the subspace \(H_{0}^{\lambda }(\Omega )\) of functions in Hölder space H λ(Ω) vanishing on ∂Ω, into the space H λ+α(Ω), if λ+α≤1. This is proved in a more general setting of generalized Hölder spaces with a given dominant of continuity modulus. Statements of such a kind are known for instance for the whole space ℝn or more generally for metric measure spaces with cancellation property. In the case of domains in ℝn when the cancellation property fails, our proofs are based on a special treatment of potential of a constant function.

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Authors and Affiliations

  1. University of Freiburg, Freiburg, Germany

    Lars Diening

  2. University of Algarve, Faro, Portugal

    Stefan G. Samko

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  1. Lars Diening
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  2. Stefan G. Samko
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Correspondence to Stefan G. Samko.

Additional information

S.G. Samko is supported by Russian Federal Targeted Programme “Scientific and Research-Educational Personnel of Innovative Russia” for 2009–2013, project N 02.740.11.5024.

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Diening, L., Samko, S.G. On potentials in generalized Hölder spaces over uniform domains in ℝn . Rev Mat Complut 24, 357–373 (2011). https://doi.org/10.1007/s13163-010-0043-6

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  • DOI: https://doi.org/10.1007/s13163-010-0043-6

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