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1
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Radial growth and variation of univalent functions and of Dirichlet finite holomorphic functions

100%
Girela D.
Colloquium Mathematicum
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1996
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tom 69
|
nr 1
19-17
EN
A well known result of Beurling asserts that if f is a function which is analytic in the unit disc $Δ ={z ∈ ℂ : |z|<1} $ and if either f is univalent or f has a finite Dirichlet integral then the set of points $e^{iθ}$ for which the radial variation $V(f,e^{iθ})=∫_{0}^{1}|f'(re^{iθ})|dr$ is infinite is a set of logarithmic capacity zero. In this paper we prove that this result is sharp in a very strong sense. Also, we prove that if f is as above then the set of points $e^{iθ}$ such that $(1 - r)|f'(re^{iθ})| ≠ o(1)$ as r → 1 is a set of logarithmic capacity zero. In particular, our results give an answer to a question raised by T. H. MacGregor in 1983.
2
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An energy estimate for the complex Monge-Ampère operator

100%
Cegrell U., Persson L.
Annales Polonici Mathematici
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1997
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tom 67
|
nr 1
95-102
EN
We prove an energy estimate for the complex Monge-Ampère operator, and a comparison theorem for the corresponding capacity and energy. The results are pluricomplex counterparts to results in classical potential theory.
3
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Musielak−Orlicz−Sobolev spaces on arbitrary metrique space

100%
Youssef A., Aissaoui N., Cherif Hassib M.
Commentationes Mathematicae
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2016
|
tom 56
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nr 2
EN
In this article we define Musielak−Orlicz−Sobolev spaces on arbitrary metric spaces with finite diameter and equipped with finite, positive Borel regular outer measure. We employ a Hajlasz definition, which uses a pointwise maximal inequality. We prove that these spaces are Banach, that the Poincaré inequality holds, and that the Lipschitz functions are dense. We develop a capacity theory based on these spaces. We study basic properties of capacity and several convergence results. As an application, we prove that each Musielak−Orlicz−Sobolev function has a quasi-continuous representative.
4
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Equality cases for condenser capacity inequalities under symmetrization

75%
Betsakos D., Pouliasis S.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2012
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tom 66
|
nr 2
EN
It is well known that certain transformations decrease the capacity of a condenser. We prove equality statements for the condenser capacity inequalities under symmetrization and polarization without connectivity restrictions on the condenser and without regularity assumptions on the boundary of the condenser.
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