Wiener's type regularity criteria on the complex plane

Józef Siciak

Annales Polonici Mathematici (1997)

  • Volume: 66, Issue: 1, page 203-221
  • ISSN: 0066-2216

Abstract

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We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that d ( t - 1 E ∩ | z - a | ≤ 1 ) ≥ c o n s t > 0 for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous.

How to cite

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Józef Siciak. "Wiener's type regularity criteria on the complex plane." Annales Polonici Mathematici 66.1 (1997): 203-221. <http://eudml.org/doc/269964>.

@article{JózefSiciak1997,
abstract = {We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that $d(t^\{-1\}E ∩ \{|z-a| ≤ 1\}) ≥ const > 0$ for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous.},
author = {Józef Siciak},
journal = {Annales Polonici Mathematici},
keywords = {subharmonic functions; logarithmic potential theory; Green function; regular points; Hölder Continuity Property; regular with respect to Dirichlet's problem; Hölder continuity of Green’s function},
language = {eng},
number = {1},
pages = {203-221},
title = {Wiener's type regularity criteria on the complex plane},
url = {http://eudml.org/doc/269964},
volume = {66},
year = {1997},
}

TY - JOUR
AU - Józef Siciak
TI - Wiener's type regularity criteria on the complex plane
JO - Annales Polonici Mathematici
PY - 1997
VL - 66
IS - 1
SP - 203
EP - 221
AB - We present a number of Wiener’s type necessary and sufficient conditions (in terms of divergence of integrals or series involving a condenser capacity) for a compact set E ⊂ ℂ to be regular with respect to the Dirichlet problem. The same capacity is used to give a simple proof of the following known theorem [2, 6]: If E is a compact subset of ℂ such that $d(t^{-1}E ∩ {|z-a| ≤ 1}) ≥ const > 0$ for 0 < t ≤ 1 and a ∈ E, where d(F) is the logarithmic capacity of F, then the Green function of ℂ E with pole at infinity is Hölder continuous.
LA - eng
KW - subharmonic functions; logarithmic potential theory; Green function; regular points; Hölder Continuity Property; regular with respect to Dirichlet's problem; Hölder continuity of Green’s function
UR - http://eudml.org/doc/269964
ER -

References

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