Displaying similar documents to “Existence theorem for n capacities”

Thin sets in nonlinear potential theory

Lars-Inge Hedberg, Thomas H. Wolff (1983)

Annales de l'institut Fourier

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Let L α q ( R D ) , α > 0 , 1 < q < ∞ , denote the space of Bessel potentials f = G α * g , g ∈ L q , with norm ∥ f ∥ α , q = ∥ g ∥ q . For α integer L α q can be identified with the Sobolev space H α , q . One can associate a potential theory to these spaces much in the same way as classical potential theory is associated to the space H 1 ; 2 , and a considerable part of the theory was carried over to this more general context around 1970. There were difficulties extending the theory of thin sets, however. By means of a new inequality, which characterizes the...

Wiener's type regularity criteria on the complex plane

Józef Siciak (1997)

Annales Polonici Mathematici

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We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that d ( t - 1 E ∩ | z - a | ≤ 1 ) ≥ c o n s t > 0 for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is...

Regularity properties of the equilibrium distribution

Hans Wallin (1965)

Annales de l'institut Fourier

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Soit F un sous-ensemble compact de R m ayant des points intérieurs et soit μ α F la distribution d’équilibre sur F de masse totale 1 par rapport au noyau r α - m avec 0 &lt; α &lt; 2 pour m ≥ 2 , et 0 &lt; α &lt; 1 pour m = 1 . La restriction de μ α F à l’intérieur de F est absolument continue et a pour densité f α F . On donne une formule explicite pour f α F et, pour une classe générale d’ensembles F , on démontre que f α F , définie en réalité sur un ensemble de mesure de Lebesgue nulle, croît comme la distance à la frontière ∂ F de F élevée à la puissance...

Potentials with respect to the pluricomplex Green function

Urban Cegrell (2012)

Annales Polonici Mathematici

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For μ a positive measure, we estimate the pluricomplex potential of μ, P μ ( x ) = ∫ Ω g ( x , y ) d μ ( y ) , where g(x,y) is the pluricomplex Green function (relative to Ω) with pole at y.

Some properties of the balayage of measures on a harmonic space

Corneliu Constantinescu (1967)

Annales de l'institut Fourier

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On démontre plusieurs théorèmes concernant le balayage des mesures sur un espace harmonique satisfaisant aux axiomes de Bauer, parmi lesquels nous indiquons les suivants : a) la balayée μ A ∪ B d’une mesure μ sur la réunion μ A ∨ μ B (dans l’espace de Riesz de mesure) ; b) ϵ x A ≠ ϵ x caractérise l’effilement de A en x  ; c) il existe un potentiel fini et continue p tel que pour tout ensemble A { x | R ^ p ( x ) &lt; p ( x ) } est exactement l’ensemble des points où A est effilé ; d) μ A est portée par la fermeture fine de A  ; e) si A et B sont...

Topological countability in Brelot potential theory

Thomas E. Armstrong (1974)

Annales de l'institut Fourier

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Let U be a domain of type H in a Brelot potential theory. A compact K in U is a G δ in U iff U - K has at most countably many components. If F is a relatively closed locally polar subset of U , any G δ in F is a G δ in U . If V is a domain in U , all Borel subsets of ∂ V ∩ U are Baire even if ∂ V ∩ U is not metrizable. The known results concerning equivalences between weak thinness, thinness, and strong thinness of a set A at a point x ∉ A are extended from the case where { x } is a G δ to the cases in which A meets only...