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

D. L. Burkholder; Richard F. Gundy

Displaying similar documents to “Boundary behaviour of harmonic functions in a half-space and brownian motion”

A non probabilistic proof of the relative Fatou theorem

J. L. Doob (1959)

Annales de l'institut Fourier

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L’auteur démontre en s’appuyant sur la thèse de Mlle Naïm le résultat suivant qu’il avait établi grâce aux probabilités : dans un espace de Green, si $u$ et $h$ sont surharmoniques $> 0$ , $u / h$ admet en tout point de l’espace ou de sa frontière de Martin une “limite fine” finie, sauf sur un ensemble de mesure nulle pour la mesure associée canoniquement à $h$ . Puis, il peut même affaiblir l’hypothèse $u > 0$ .

On the boundary limits of harmonic functions with gradient in $L^{p}$

Yoshihiro Mizuta (1984)

Annales de l'institut Fourier

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This paper deals with tangential boundary behaviors of harmonic functions with gradient in Lebesgue classes. Our aim is to extend a recent result of Cruzeiro (C.R.A.S., Paris, 294 (1982), 71–74), concerning tangential boundary limits of harmonic functions with gradient in $L^{n} (R_{+}^{n})$ , $R_{+}^{n}$ denoting the upper half space of the $n$ -dimensional euclidean space $R^{n}$ . Our method used here is different from that of Nagel, Rudin and Shapiro (Ann. of Math., 116 (1982), 331–360); in fact, we use the integral representation...

On separately subharmonic functions (Lelong’s problem)

A. Sadullaev (2011)

Annales de la faculté des sciences de Toulouse Mathématiques

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The main result of the present paper is : every separately-subharmonic function $u (x, y)$ , which is harmonic in $y$ , can be represented locally as a sum two functions, $u = u^{*} + U$ , where $U$ is subharmonic and $u^{*}$ is harmonic in $y$ , subharmonic in $x$ and harmonic in $(x, y)$ outside of some nowhere dense set $S$ .

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

Alexey Karapetyants, Joel Esteban Restrepo (2020)

Czechoslovak Mathematical Journal

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We study spaces of Hölder type functions harmonic in the unit ball and half space with some smoothness conditions up to the boundary. The first type is the Hölder type space of harmonic functions with prescribed modulus of continuity $ω = ω (h)$ and the second is the variable exponent harmonic Hölder space with the continuity modulus ${| h |}^{λ (\cdot)}$ . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

$p$ -harmonic measure is not additive on null sets

José G. Llorente, Juan J. Manfredi, Jang-Mei Wu (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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When $1 < p < \infty$ and $p \neq 2$ the $p$ -harmonic measure on the boundary of the half plane $ℝ_{+}^{2}$ is not additive on null sets. In fact, there are finitely many sets $E_{1}$ , $E_{2}$ ,..., $E_{κ}$ in $ℝ$ , of $p$ -harmonic measure zero, such that $E_{1} \cup E_{2} \cup . . . \cup E_{κ} = ℝ$ .

Duality on vector-valued weighted harmonic Bergman spaces

Salvador Pérez-Esteva (1996)

Studia Mathematica

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We study the duals of the spaces $A^{p α} (X)$ of harmonic functions in the unit ball of $ℝ^{n}$ with values in a Banach space X, belonging to the Bochner $L^{p}$ space with weight ${(1 - | x |)}^{α}$ , denoted by $L^{p α} (X)$ . For 0 < α < p-1 we construct continuous projections onto $A^{p α} (X)$ providing a decomposition $L^{p α} (X) = A^{p α} (X) + M^{p α} (X)$ . We discuss the conditions on p, α and X for which $A^{p α} (X) * = A^{q α} (X *)$ and $M^{p α} (X) * = M^{q α} (X *)$ , 1/p+1/q = 1. The last equality is equivalent to the Radon-Nikodým property of X*.

Displaying similar documents to “Boundary behaviour of harmonic functions in a half-space and brownian motion”

A non probabilistic proof of the relative Fatou theorem

On the boundary limits of harmonic functions with gradient in L p

On separately subharmonic functions (Lelong’s problem)

Generalized Hölder type spaces of harmonic functions in the unit ball and half space

p -harmonic measure is not additive on null sets

Duality on vector-valued weighted harmonic Bergman spaces

On the boundary limits of harmonic functions with gradient in $L^{p}$

$p$ -harmonic measure is not additive on null sets