Logarithmic capacity is not subadditive – a fine topology approach

Pavel Pyrih

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 1, page 67-72
  • ISSN: 0010-2628

Abstract

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In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.

How to cite

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Pyrih, Pavel. "Logarithmic capacity is not subadditive – a fine topology approach." Commentationes Mathematicae Universitatis Carolinae 33.1 (1992): 67-72. <http://eudml.org/doc/247386>.

@article{Pyrih1992,
abstract = {In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.},
author = {Pyrih, Pavel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {logarithmic capacity; fine topology; Choquet capacity; subadditivity; Wiener's test; logarithmic capacity},
language = {eng},
number = {1},
pages = {67-72},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Logarithmic capacity is not subadditive – a fine topology approach},
url = {http://eudml.org/doc/247386},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Pyrih, Pavel
TI - Logarithmic capacity is not subadditive – a fine topology approach
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 1
SP - 67
EP - 72
AB - In Landkof’s monograph [8, p. 213] it is asserted that logarithmic capacity is strongly subadditive, and therefore that it is a Choquet capacity. An example demonstrating that logarithmic capacity is not even subadditive can be found e.gi̇n [6, Example 7.20], see also [3, p. 803]. In this paper we will show this fact with the help of the fine topology in potential theory.
LA - eng
KW - logarithmic capacity; fine topology; Choquet capacity; subadditivity; Wiener's test; logarithmic capacity
UR - http://eudml.org/doc/247386
ER -

References

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  1. Brelot M., Lectures on Potential Theory, Tata Institute of Fundamental Research Bombay (1966). (1966) MR0118980
  2. Brelot M., On Topologies and Boundaries in Potential Theory, Lecture Notes in Mathematics No. 175, Springer-Verlag, Berlin (1971). (1971) Zbl0222.31014MR0281940
  3. Doob J.L., Classical Potential Theory and Its Probabilistic Counterpart, Springer, New-York (1984). (1984) Zbl0549.31001MR0731258
  4. Fuglede B., Fine Topology and Finely Holomorphic Functions, Proc. 18th Scand. Congr. Math. Aarhus (1980), 22-38. (1980) MR0633349
  5. Fuglede B., Sur les fonctions finement holomorphes, Ann. Inst. Fourier (Grenoble) 31.4 (1981), 57-88. (1981) Zbl0445.30040MR0644343
  6. Hayman W.K., Subharmonic functions, Vol.2, London Math. Society Monographs 20, Academic Press London (1989). (1989) MR1049148
  7. Helms L.L., Introduction to Potential Theory, Wiley Interscience Pure and Applied Mathematics 22, New-York (1969). (1969) Zbl0188.17203MR0261018
  8. Landkof N.S., Foundations of Modern Potential Theory, Russian Moscow (1966). (1966) 
  9. Landkof N.S., Foundations of Modern Potential Theory, (translation from [8]), Springer-Verlag, Berlin (1972). (1972) Zbl0253.31001MR0350027
  10. Lukeš J., Malý J., Zajíček L., Fine Topology Methods in Real Analysis and Potential Theory, Lecture Notes in Mathematics No. 1189, Springer-Verlag, Berlin (1986). (1986) MR0861411

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