Riv. Mat. Univ. Parma, Vol. 10, No. 1, 2019

Mohamed El Kadiri [a], Abderrahim Aslimani [b] and Sabah Haddad [c]

On the integral representation of the nonnegative superharmonic functions in a balayage space

Pages: 1-24
Received: 6 July 2018
Accepted: 7 May 2019
Mathematics Subject Classification (2010): 31B05, 31B10, 31C35, 31C99.
Keywords: Balayage space, Green function, potential, axiom of proportionality, extreme element, integral representation.
Authors address:
[a],[b]: University of Mohammed V, Dept. of Mathematics, Faculty of Sciences, P.B. 1014, Rabat, Morocco
[c]: CRMEF, Rabat-Akkari, Morocco

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Abstract: In this paper we study the integral representation of nonnegative superharmonic functions in a balayage space \((X,\mathcal{W})\) by using Choquet's method. When the space \(X\) has a Green kernel \(G\), we show that if a sequence of potentials in \(X\) are representable by \(G\) and majorized by some potential converges in the natural topology to a superharmonic function \(p\) on \(X\), then \(p\) is representable by \(G\). If in addition of the existence of the Green kernel, the potentials of harmonic support reduced to a single point are proportional, then any potential on \(X\) can be represented by the function \(G\) and reciprocally.

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