A general definition of capacity

Makoto Ohtsuka

Displaying similar documents to “A general definition of capacity”

Thin sets in nonlinear potential theory

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

Annales de l'institut Fourier

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Let $L_{α}^{q} (R^{D}), α > 0, 1 < q < \infty$ , denote the space of Bessel potentials $f = G_{α} * g$ , $g \in L^{q}$ , 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...

A new setting for potential theory. I

Kai Lai Chung, K. Murali Rao (1980)

Annales de l'institut Fourier

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We consider a transient Hunt process in which the potential density $u$ satisfies the conditions: (a) for each $x$ , $u (x, y)^{- 1}$ is finite continuous in $y$ ; (b) $u (x, y) = + \infty$ iff $x = y$ . In earlier papers Chung established an equilibrium principle, and Rao obtained a Riesz of decomposition for excessive functions. We now begin a deeper study under these conditions, including the uniqueness of the decomposition and Hunt’s hypothesis (B).

Negligible sets and good functions on polydiscs

Kohur Gowrisankaran (1979)

Annales de l'institut Fourier

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A notion of negligible sets for polydiscs is introduced. Some properties of non-negligible sets are proved. These results are used to construct good and good inner functions on polydiscs.

The Wiener test for degenerate elliptic equations

E. B. Fabes, D. S. Jerison, C. E. Kenig (1982)

Annales de l'institut Fourier

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We consider degenerated elliptic equations of the form $\sum_{i, j} D_{x_{i}} (a_{i j} (x) D_{x_{j}}), where λ w (x) | ξ |^{2} \leq \sum_{i, j} a_{i j} (x) ξ_{i} ξ_{j} \leq Λ w (x) | ξ |^{2} .$ Under suitable assumptions on $w$ , we obtain a characterization of Wiener type (involving weighted capacities) for the set of regular points for these operators. The set of regular points is shown to depend only on $w$ . The main tool we use is an estimate for the Green function in terms of $w$ .

Integral representation for a class of multiply superharmonic functions

Kohur Gowrisankaran (1973)

Annales de l'institut Fourier

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Let $Ω_{1}, ..., Ω_{n}$ be harmonic spaces of Brelot with countable base of completely determining domains. The elements of a subcone $C$ of the cone of positive $n$ -superharmonic functions in $Ω_{1} \times ... \times Ω_{n}$ is shown to have an integral representation with the aid of Radon measures on the extreme elements belonging to a compact base of $C$ . The extreme elements are shown to be the product of extreme superharmonic functions on the component spaces and the measure representing each element is shown to be unique. Necessary...