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Capacities on Harmonic Spaces with Adjoint Structure

In: Potential Theory

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  • Fumi-Yuki Maeda

    (Hiroshima University, Department of Mathematics, Faculty of Science)

Abstract

In the classical potential theory, the capacities defined in terms of Green potentials coincide with the capacity defined by Dirichlet integrals; more precisely, for a compact set K in a Greenian domain Ω in ℝd, $$Sup\{ \mu (\Omega )|G\mu \leqq 1,\,Supp\,\mu \subset K\} = \inf \{ \smallint G\mu \,d\mu |G\mu \geqq 1\,on\,K\} = \,\inf \{ D[f]\,|\,f:\,potential\,on\,\Omega \,with\,f \geqq 1\,on\,K\}$$ , where Gμ is the Green potential of μ ≧ 0 on Ω and D[f] is the Dirichlet integral of f.

Suggested Citation

  • Fumi-Yuki Maeda, 1988. "Capacities on Harmonic Spaces with Adjoint Structure," Springer Books, in: Josef Král & Jaroslav Lukeš & Ivan Netuka & Jiří Veselý (ed.), Potential Theory, pages 231-236, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4613-0981-9_30
    DOI: 10.1007/978-1-4613-0981-9_30
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