On the Dirichlet problem for functions of the first Baire class

Jiří Spurný

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 4, page 721-728
  • ISSN: 0010-2628

Abstract

top
Let ℋ be a simplicial function space on a metric compact space X . Then the Choquet boundary Ch X of ℋ is an F σ -set if and only if given any bounded Baire-one function f on Ch X there is an ℋ -affine bounded Baire-one function h on X such that h = f on Ch X . This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set X .

How to cite

top

Spurný, Jiří. "On the Dirichlet problem for functions of the first Baire class." Commentationes Mathematicae Universitatis Carolinae 42.4 (2001): 721-728. <http://eudml.org/doc/248820>.

@article{Spurný2001,
abstract = {Let $\mathcal \{H\}$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname\{Ch\}X$ of $\mathcal \{H\}$ is an $F_\sigma $-set if and only if given any bounded Baire-one function $f$ on $\operatorname\{Ch\}X$ there is an $\mathcal \{H\}$-affine bounded Baire-one function $h$ on $X$ such that $h=f$ on $\operatorname\{Ch\}X$. This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$.},
author = {Spurný, Jiří},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions; weak Dirichlet problem; function space; Choquet simplex; Baire-one function},
language = {eng},
number = {4},
pages = {721-728},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the Dirichlet problem for functions of the first Baire class},
url = {http://eudml.org/doc/248820},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Spurný, Jiří
TI - On the Dirichlet problem for functions of the first Baire class
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 4
SP - 721
EP - 728
AB - Let $\mathcal {H}$ be a simplicial function space on a metric compact space $X$. Then the Choquet boundary $\operatorname{Ch}X$ of $\mathcal {H}$ is an $F_\sigma $-set if and only if given any bounded Baire-one function $f$ on $\operatorname{Ch}X$ there is an $\mathcal {H}$-affine bounded Baire-one function $h$ on $X$ such that $h=f$ on $\operatorname{Ch}X$. This theorem yields an answer to a problem of F. Jellett from [8] in the case of a metrizable set $X$.
LA - eng
KW - weak Dirichlet problem; function space; Choquet simplexes; Baire-one functions; weak Dirichlet problem; function space; Choquet simplex; Baire-one function
UR - http://eudml.org/doc/248820
ER -

References

top
  1. Alfsen E.M., Compact convex sets and boundary integrals, Springer-Verlag New York-Heidelberg (1971). (1971) Zbl0209.42601MR0445271
  2. Bauer H., Axiomatische behandlung des Dirichletschen problems fur elliptische und parabolische differentialgleichungen, Math. Ann. 146 (1962), 1-59. (1962) MR0144064
  3. Boboc N., Cornea A., Convex cones of lower semicontinuous functions on compact spaces, Rev. Roum. Math. Pures. App. 12 (1967), 471-525. (1967) Zbl0155.17301MR0216278
  4. Bliedtner J., Hansen W., Simplicial cones in potential theory, Invent. Math. (2) 29 (1975), 83-110. (1975) Zbl0308.31011MR0387630
  5. Capon M., Sur les fonctions qui vérifient le calcul barycentrique, Proc. London Math. Soc. (3) 32 (1976), 163-180. (1976) Zbl0313.46003MR0394148
  6. Engelking R., General Topology, Heldermann, Berlin (1989). (1989) Zbl0684.54001MR1039321
  7. Choquet G., Lectures on analysis vol. II: Representation theory, W.A. Benjamin, Inc., New York-Amsterdam (1969). (1969) Zbl0181.39602MR0250012
  8. Jellett F., On affine extensions of continuous functions defined on the extreme boundary of a Choquet simplex, Quart. J. Math. Oxford (2) 36 (1985), 71-73. (1985) Zbl0582.46010MR0780351
  9. Lacey H.E.. Morris P.D., On spaces of type A ( K ) and their duals, Proc. Amer. Math. Soc. 23 (1969), 151-157. (1969) MR0625855
  10. Lukeš J., Malý J., Zajíček L., Fine topology methods in real analysis and potential theory, Lecture Notes in Math. 1189 Springer-Verlag (1986). (1986) MR0861411
  11. Phelps R.R., Lectures on Choquet's theorem, D. Van Nostrand Co. (1966). (1966) Zbl0135.36203MR0193470

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.