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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.

Behaviour at the boundary of the complex Monge-Ampère equation

J. Lee, R. Melrose (1980/1981)

Séminaire Équations aux dérivées partielles (Polytechnique)

Beiträge zum potentialtheoretischen Aspekt des Verheftungssatzes von A.D. Alexandrow.

Heinz Leutwiler (1970)

Commentarii mathematici Helvetici

Bernstein Theorems for Harmonic Morphisms from R3 and S3.

Paul Baird, John C. Wood (1988)

Mathematische Annalen

Besov capacity and Hausdorff measures in metric measure spaces.

S. Costea (2009)

Publicacions Matemàtiques

Besov Regularity for Second Order Elliptic Boundary Value Problems with Variable Coefficients.

Stephan Dahlke (1998)

Manuscripta mathematica

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.

Best constants for some operators associated with the Fourier and Hilbert transforms

B. Hollenbeck, N. J. Kalton, I. E. Verbitsky (2003)

Studia Mathematica

We determine the norm in $L^{p} (ℝ ₊)$ , 1 < p < ∞, of the operator $I - ℱ_{s} ℱ_{c}$ , where $ℱ_{c}$ and $ℱ_{s}$ are respectively the cosine and sine Fourier transforms on the positive real axis, and I is the identity operator. This solves a problem posed in 1984 by M. S. Birman [Bir] which originated in scattering theory for unbounded obstacles in the plane. We also obtain the $L^{p}$ -norms of the operators aI + bH, where H is the Hilbert transform (conjugate function operator) on the circle or real line, for arbitrary real a,b. Best...

Best constants in the Harnack inequality for the Weinstein equation.

Heinz Leutwiler (1987)

Aequationes mathematicae

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 Green's functions and biharmonic degeneracy.

C.Y. Wang (1975)

Mathematica Scandinavica

Biharmonic Green's Functions and Quasiharmonic Degeneracy.

C.Y. Wang (1974)

Mathematica Scandinavica

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.

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