Comptes Rendus
Mathematical Analysis/Harmonic Analysis
On the regular convergence of multiple integrals of locally Lebesgue integrable functions over R¯+m
Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 459-464.

Let the function f:R¯+mC be such that fLloc1(R¯+m), where m2 is a fixed integer. We investigate the convergence behavior of the m-multiple integral

0v10v20vmf(t1,t2,,tm)dt1dt2dtmas min{v1,v2,,vm},
while using two notions of convergence: the one in Pringsheimʼs sense and the one in the regular sense. For the sake of brevity, we present our main result in the case m=2 as follows: If fLloc1(R¯+2) and the double integral (⁎) converges regularly, then the finite limits limv20v1(0v2f(t1,t2)dt2)dt1=:J1(v1) and limv10v2(0v1f(t1,t2)dt1)dt2=:J2(v2) exist uniformly in 0<v1,v2<, respectively, and limv1J1(v1)=limv2J2(v2)=limv1,v20v10v2f(t1,t2)dt1dt2. This can be considered as a generalized version of Fubiniʼs theorem on successive integration in the case when fL1(R¯+2).

Soit f:R¯+mC telle que fLloc1(R¯+m), où m est un entier fixé. On étudie la convergence de lʼintégrale multiple dʼordre m, 0v10v20vmf(t1,t2,,tm)dt1dt2dtm quand min{v1,v2,,vm}, en utilisant deux méthodes de convergence, lʼune au sens de Pringsheim, et lʼautre au sens régulier. Pour simplifier on présente notre résultat fondamental pour m=2, de la façon suivante : Si fLloc1(R¯+2) et si lʼintégrale double converge régulièrement, alors les limites finies limv20v1(0v2f(t1,t2)dt2)dt1=:J1(v1) et limv10v2(0v1f(t1,t2)dt1)dt2=:J2(v2) existent uniformément dans 0<v1,v2<, respectivement, et on a limv1J1(v1)=limv2J2(v2)=limv1,v20v10v2f(t1,t2)dt1dt2. Ceci peut être considéré comme une généralisation du théorème de Fubini concernant lʼintégration successive au cas où fL1(R¯+2).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2012.03.010

Ferenc Móricz 1

1 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary
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Ferenc Móricz. On the regular convergence of multiple integrals of locally Lebesgue integrable functions over $ {\overline{\mathbb{R}}}_{+}^{m}$. Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 459-464. doi : 10.1016/j.crma.2012.03.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.03.010/

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