# Wiener's type regularity criteria on the complex plane

Annales Polonici Mathematici (1997)

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

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topJózef 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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