Displaying similar documents to “On the boundary limits of harmonic functions with gradient in L p ”

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 < ∞ and p ≠ 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 ∪ E 2 ∪ . . . ∪ E κ = ℝ .

Hörmander systems and harmonic morphisms

Elisabetta Barletta (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Given a Hörmander system X = { X 1 , ⋯ , X m } on a domain Ω ⊆ 𝐑 n we show that any subelliptic harmonic morphism φ from Ω into a ν -dimensional riemannian manifold N is a (smooth) subelliptic harmonic map (in the sense of J. Jost & C-J. Xu, [9]). Also φ is a submersion provided that ν ≤ m and X has rank m . If Ω = 𝐇 n (the Heisenberg group) and X = 1 2 L α + L α ¯ , 1 2 i L α - L α ¯ , where L α ¯ = ∂ / ∂ z ¯ α - i z α ∂ / ∂ t is the Lewy operator, then a smooth map φ : Ω → N is a subelliptic harmonic morphism if and only if φ ∘ π : ( C ( 𝐇 n ) , F θ 0 ) → N is a harmonic morphism, where S 1 → C ( 𝐇 n ) → π → 𝐇 n is the canonical circle bundle and F θ 0 ...

On Φ -bounded harmonic functions

Mitsuru Nakai (1966)

Annales de l'institut Fourier

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Soit Φ une fonction non négative réelle ; une fonction u harmonique sur une surface de Riemann R est dite Φ -bornée si Φ ( | u | ) admet une majorante harmonique. On étudie la classe H Φ ( R ) des fonctions Φ -bornées sur R et on montre, en particulier, que chaque u de H Φ ( R ) est essentiellement positive pour toute R , si et seulement si inf t → ∞ Φ ( t ) / t > 0 .

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 ( · ) . We give a characterization of functions in these spaces in terms of the behavior of their derivatives near the boundary.

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 .

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

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

Annales de l'institut Fourier

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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...