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Determination of the pluripolar hull of graphs of certain holomorphic functions

Armen Edigarian, Jan Wiegerinck (2004)

Annales de l’institut Fourier

Let A be a closed polar subset of a domain D in ℂ . We give a complete description of the pluripolar hull Γ D × ℂ * of the graph Γ of a holomorphic function defined on D ∖ A . To achieve this, we prove for pluriharmonic measure certain semi-continuity properties and a localization principle.

Dichotomy of global density of Riesz capacity

Hiroaki Aikawa (2016)

Studia Mathematica

Let C α be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density ̲ ( C α , E , r ) = i n f x ∈ ℝ ⁿ C α ( E ∩ B ( x , r ) ) / C α ( B ( x , r ) ) for a Borel set E ⊂ ℝⁿ, where B(x,r) stands for the open ball with center at x and radius r. In case 0 < α ≤ 2, we show that l i m r → ∞ ̲ ( C α , E , r ) is either 0 or 1; the first case occurs if and only if ̲ ( C α , E , r ) is identically zero for all r > 0. Moreover, it is shown that the densities with respect to more general open sets enjoy the same dichotomy. A decay estimate for α-capacitary potentials is also obtained.

Dirichlet problem with L p -boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let ℒ be a sub-laplacian on a stratified Lie group G . In this paper we study the Dirichlet problem for ℒ with L p -boundary data, on domains Ω which are contractible with respect to the natural dilations of G . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for ℒ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

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