On the integral representation of finely superharmonic functions

Abderrahim Aslimani; Imad El Ghazi; Mohamed El Kadiri

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 3, page 323-350
  • ISSN: 0010-2628

Abstract

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In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset U of a Brelot 𝒫 -harmonic space Ω with countable base of open subsets and satisfying the axiom D . When Ω satisfies the hypothesis of uniqueness, we define the Martin boundary of U and the Martin kernel K and we obtain the integral representation of invariant functions by using the kernel K . As an application of the integral representation we extend to the cone 𝒮 ( 𝒰 ) of nonnegative finely superharmonic functions in U a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.

How to cite

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Aslimani, Abderrahim, El Ghazi, Imad, and El Kadiri, Mohamed. "On the integral representation of finely superharmonic functions." Commentationes Mathematicae Universitatis Carolinae 60.3 (2019): 323-350. <http://eudml.org/doc/294537>.

@article{Aslimani2019,
abstract = {In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $\mathcal \{P\}$-harmonic space $\Omega $ with countable base of open subsets and satisfying the axiom $D$. When $\Omega $ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $\mathcal \{S(U)\}$ of nonnegative finely superharmonic functions in $U$ a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.},
author = {Aslimani, Abderrahim, El Ghazi, Imad, El Kadiri, Mohamed},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {finely harmonic function; finely superharmonic function; fine potential; fine Green kernel; integral representation; Martin boundary; fine Riesz-Martin kernel},
language = {eng},
number = {3},
pages = {323-350},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On the integral representation of finely superharmonic functions},
url = {http://eudml.org/doc/294537},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Aslimani, Abderrahim
AU - El Ghazi, Imad
AU - El Kadiri, Mohamed
TI - On the integral representation of finely superharmonic functions
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 3
SP - 323
EP - 350
AB - In the present paper we study the integral representation of nonnegative finely superharmonic functions in a fine domain subset $U$ of a Brelot $\mathcal {P}$-harmonic space $\Omega $ with countable base of open subsets and satisfying the axiom $D$. When $\Omega $ satisfies the hypothesis of uniqueness, we define the Martin boundary of $U$ and the Martin kernel $K$ and we obtain the integral representation of invariant functions by using the kernel $K$. As an application of the integral representation we extend to the cone $\mathcal {S(U)}$ of nonnegative finely superharmonic functions in $U$ a partition theorem of Brelot. We also establish an approximation result of invariant functions by finely harmonic functions in the case where the minimal invariant functions are finely harmonic.
LA - eng
KW - finely harmonic function; finely superharmonic function; fine potential; fine Green kernel; integral representation; Martin boundary; fine Riesz-Martin kernel
UR - http://eudml.org/doc/294537
ER -

References

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