Displaying similar documents to “A new setting for potential theory. I”

Analytic potential theory over the p -adics

Shai Haran (1993)

Annales de l'institut Fourier

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Over a non-archimedean local field the absolute value, raised to any positive power α > 0 , is a negative definite function and generates (the analogue of) the symmetric stable process. For α ∈ ( 0 , 1 ) , this process is transient with potential operator given by M. Riesz’ kernel. We develop this potential theory purely analytically and in an explicit manner, obtaining special features afforded by the non-archimedean setting ; e.g. Harnack’s inequality becomes an equality.

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...

Duality and the Martin compactification

John C. Taylor (1972)

Annales de l'institut Fourier

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Let ℋ be a Bauer sheaf that admits a Green function. Then there exists a diffusion process corresponding to the sheaf whose resolvent possesses a Hunt-Kunita-Watanabe dual resolvent that comes from a diffusion process. If ℋ is a Brelot sheaf which possesses an adjoint sheaf ℋ * the dual process corresponds to ℋ * . The Martin compactification defined by a Brelot sheaf ℋ that admits a Green function coincides with a Kunita-Watanabe compactification defined by the dual resolvent. ...

Mean values and associated measures of δ -subharmonic functions

Neil A. Watson (2002)

Mathematica Bohemica

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Let u be a δ -subharmonic function with associated measure μ , and let v be a superharmonic function with associated measure ν , on an open set E . For any closed ball B ( x , r ) , of centre x and radius r , contained in E , let ℳ ( u , x , r ) denote the mean value of u over the surface of the ball. We prove that the upper and lower limits as s , t → 0 with 0 < s < t of the quotient ( ℳ ( u , x , s ) - ℳ ( u , x , t ) ) / ( ℳ ( v , x , s ) - ℳ ( v , x , t ) ) , lie between the upper and lower limits as r → 0 + of the quotient μ ( B ( x , r ) ) / ν ( B ( x , r ) ) . This enables us to use some well-known measure-theoretic results to prove new variants...