Geodesic
On the integral representation of finely superharmonic functions
Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 3, pp. 323-350.

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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 $\mathcal{P}$-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.
DOI : 10.14712/1213-7243.2019.019
Classification : 31C35, 31C40, 31D05
Keywords: finely harmonic function; finely superharmonic function; fine potential; fine Green kernel; integral representation; Martin boundary; fine Riesz-Martin kernel 
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Aslimani, Abderrahim; El Ghazi, Imad; El Kadiri, Mohamed. On the integral representation of finely superharmonic functions. Commentationes Mathematicae Universitatis Carolinae, Tome 60 (2019) no. 3, pp. 323-350. doi : 10.14712/1213-7243.2019.019. http://geodesic.mathdoc.fr/articles/10.14712/1213-7243.2019.019/

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