Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-22T20:41:23.606Z Has data issue: false hasContentIssue false
  • English
  • Français

THE DIRICHLET PROBLEM FOR BAIRE-TWO FUNCTIONS ON SIMPLICES

Part of: Topological linear spaces and related structures

Published online by Cambridge University Press:  26 February 2009

JIŘÍ SPURNÝ
JIŘÍ SPURNÝ*
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that solvability of the abstract Dirichlet problem for Baire-two functions on a simplex X cannot be characterized by topological properties of the set of extreme points of X.

Keywords

simplexstrongly affine functionsbarycentric formulaBaire-two functions

MSC classification

Secondary: 46A55: Convex sets in topological linear spaces; Choquet theory
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 79 , Issue 2 , April 2009 , pp. 285 - 297
Copyright
Copyright © Australian Mathematical Society 2009

Footnotes

The work is a part of the research project MSM 0021620839 financed by MSMT and partly supported by the grants GA ČR 201/06/0018 and GA ČR 201/07/0388.

References

[1] Bauer, H., ‘Šilowscher Rand und Dirichletsches Problem’, Ann. Inst. Fourier (Grenoble) 11 (1961), 89–136.CrossRefGoogle Scholar
[2] Bliedtner, J. and Hansen, W., Potential Theory – an Analytic and Probabilistic Approach to Balayage (Springer, Berlin, 1986).CrossRefGoogle Scholar
[3] Choquet, G., ‘Remarques à propos de la démonstration d’unicité de P.A. Meyer’, Séminaire Brelot-Choquet-Deny. Théorie du potentiel 6 (1961–1962), 1–13.Google Scholar
[4] Engelking, R., General Topology (Heldermann Verlag, Berlin, 1989).Google Scholar
[5] Fonf, V. P., Lindenstrauss, J. and Phelps, R. R., ‘Infinite dimensional convexity’, in: Handbook of the Geometry of Banach Spaces Vol. I (North-Holland, Amsterdam, 2001), pp. 599–670.CrossRefGoogle Scholar
[6] Holický, P., Zajíček, L. and Zelený, M., ‘A remark on a theorem of Solecki’, Comment. Math. Univ. Carolin. 46(1) (2005), 43–54.Google Scholar
[7] Kechris, A. S., Classical Descriptive Set Theory (Springer, New York, 1995).CrossRefGoogle Scholar
[8] Krause, U., ‘Der Satz von Choquet als ein abstrakter Spektralsatz und vice versa’, Math. Ann. 184 (1970), 275–296.CrossRefGoogle Scholar
[9] Lukeš, J., Malý, J., Netuka, I., Smrčka, M. and Spurný, J., ‘On approximation of affine Baire-one functions’, Israel J. Math. 134 (2003), 255–289.CrossRefGoogle Scholar
[10] Lukeš, J., Mocek, T., Smrčka, M. and Spurný, J., ‘Choquet like sets in function spaces’, Bull. Sci. Math. 127 (2003), 397–437.CrossRefGoogle Scholar
[11] Lusin, N. N., Collected Works, Part 2 (Izdat. Akad. Nauk SSR, Moscow, 1958) (in Russian).Google Scholar
[12] Saint Raymond, J., ‘Fonctions convexes sur un convexe borné complet’, Bull. Sci. Math. 102(2) (1978), 331–336.Google Scholar
[13] Spurný, J., ‘Affine Baire-one functions on Choquet simplexes’, Bull. Aust. Math. Soc. 71 (2005), 235–258.CrossRefGoogle Scholar
[14] Spurný, J., Baire classes of Banach spaces and strongly affine functions, Trans. Amer. Math. Soc., Preprint: available at http://www.karlin.mff.cuni.cz/∼rokyta/preprint/index.php; to appear.Google Scholar
[15] Spurný, J., ‘Representation of abstract affine functions’, Real. Anal. Exchange 28(2) (2002/2003), 337–354.CrossRefGoogle Scholar
[16] Spurný, J. and Kalenda, O., ‘A solution of the abstract Dirichlet problem for Baire-one functions’, J. Funct. Anal. 232 (2006), 259–294.CrossRefGoogle Scholar