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Behavior of biharmonic functions on Wiener's and Royden's compactifications

Y. K. Kwon, Leo Sario, Bertram Walsh (1971)

Annales de l'institut Fourier

Let R be a smooth Riemannian manifold of finite volume, Δ its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of R are found, and for biharmonic functions (those for which Δ Δ u = 0 ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.

Bessel potentials in Orlicz spaces.

N. Aïssaoui (1997)

Revista Matemática de la Universidad Complutense de Madrid

It is shown that Bessel potentials have a representation in term of measure when the underlying space is Orlicz. A comparison between capacities and Lebesgue measure is given and geometric properties of Bessel capacities in this space are studied. Moreover it is shown that if the capacity of a set is null, then the variation of all signed measures of this set is null when these measures are in the dual of an Orlicz-Sobolev space.

Biharmonic Green domains in a Riemannian manifold

Sadoon Ibrahim Othman, Victor Anandam (2003)

Commentationes Mathematicae Universitatis Carolinae

Let R be a Riemannian manifold without a biharmonic Green function defined on it and Ω a domain in R . A necessary and sufficient condition is given for the existence of a biharmonic Green function on Ω .

Biharmonic morphisms

Mustapha Chadli, Mohamed El Kadiri, Sabah Haddad (2005)

Commentationes Mathematicae Universitatis Carolinae

Let ( X , ℋ ) and ( X ' , ℋ ' ) be two strong biharmonic spaces in the sense of Smyrnelis whose associated harmonic spaces are Brelot spaces. A biharmonic morphism from ( X , ℋ ) to ( X ' , ℋ ' ) is a continuous map from X to X ' which preserves the biharmonic structures of X and X ' . In the present work we study this notion and characterize in some cases the biharmonic morphisms between X and X ' in terms of harmonic morphisms between the harmonic spaces associated with ( X , ℋ ) and ( X ' , ℋ ' ) and the coupling kernels of them.

Boundary behavior of subharmonic functions in nontangential accessible domains

Shiying Zhao (1994)

Studia Mathematica

The following results concerning boundary behavior of subharmonic functions in the unit ball of ℝ n are generalized to nontangential accessible domains in the sense of Jerison and Kenig [7]: (i) The classical theorem of Littlewood on the radial limits. (ii) Ziomek’s theorem on the L p -nontangential limits. (iii) The localized version of the above two results and nontangential limits of Green potentials under a certain nontangential condition.

Boundary behaviour of harmonic functions in a half-space and brownian motion

D. L. Burkholder, Richard F. Gundy (1973)

Annales de l'institut Fourier

Let u be harmonic in the half-space R + n + 1 , n ≥ 2 . We show that u can have a fine limit at almost every point of the unit cubs in R n = ∂ R + n + 1 but fail to have a nontangential limit at any point of the cube. The method is probabilistic and utilizes the equivalence between conditional Brownian motion limits and fine limits at the boundary.In R + 2 it is known that the Hardy classes H p , 0 < p < ∞ , may be described in terms of the integrability of the nontangential maximal function, or, alternatively, in terms of the integrability...

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