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"Counterexamples" to the harmonic Liouville theorem and harmonic functions with zero nontangential limits

A. Bonilla (2000)

Colloquium Mathematicae

We prove that, if μ>0, then there exists a linear manifold M of harmonic functions in which is dense in the space of all harmonic functions in and lim‖x‖→∞ x ∈ S ‖x‖μDαv(x) = 0 for every v ∈ M and multi-index α, where S denotes any hyperplane strip. Moreover, every nonnull function in M is universal. In particular, if μ ≥ N+1, then every function v ∈ M satisfies ∫H vdλ =0 for every (N-1)-dimensional hyperplane H, where λ denotes the (N-1)-dimensional Lebesgue measure. On the other hand, we...

Determination of the pluripolar hull of graphs of certain holomorphic functions

Armen Edigarian, Jan Wiegerinck (2004)

Annales de l’institut Fourier

Let be a closed polar subset of a domain in . We give a complete description of the pluripolar hull of the graph of a holomorphic function defined on . 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 be the Riesz capacity of order α, 0 < α < n, in ℝⁿ. We consider the Riesz capacity density 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 is either 0 or 1; the first case occurs if and only if 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-boundary data in contractible domains of Carnot groups

Andrea Bonfiglioli, Ermanno Lanconelli (2006)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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

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