On the regular convergence of multiple integrals of locally Lebesgue integrable functions over $ {\overline{\mathbb{R}}}_{+}^{m}$

Ferenc Móricz

doi:10.1016/j.crma.2012.03.010

Mathematical Analysis/Harmonic Analysis

On the regular convergence of multiple integrals of locally Lebesgue integrable functions over

{\bar{R}}_{+}^{m}

[Sur la convergence régulière dʼintégrales multiples définies sur

{\bar{R}}_{+}^{m}

localement intégrables au sens de Lebesgue]

Présenté par : Jean-Pierre Kahane

Ferenc Móricz ¹

¹ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary

Comptes Rendus. Mathématique, Volume 350 (2012) no. 9-10, pp. 459-464.

Résumé
Abstract

Soit $f : {\bar{R}}_{+}^{m} \to C$ telle que $f \in L_{loc}^{1} ({\bar{R}}_{+}^{m})$ , où m est un entier fixé. On étudie la convergence de lʼintégrale multiple dʼordre m, $\int_{0}^{v_{1}} \int_{0}^{v_{2}} \dots \int_{0}^{v_{m}} f (t_{1}, t_{2}, \dots, t_{m}) d t_{1} d t_{2} \dots d t_{m}$ quand $\min {v_{1}, v_{2}, \dots, v_{m}} \to \infty$ , 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 $f \in L_{loc}^{1} ({\bar{R}}_{+}^{2})$ et si lʼintégrale double converge régulièrement, alors les limites finies $\lim_{v_{2} \to \infty} \int_{0}^{v_{1}} (\int_{0}^{v_{2}} f (t_{1}, t_{2}) d t_{2}) d t_{1} = : J_{1} (v_{1})$ et $\lim_{v_{1} \to \infty} \int_{0}^{v_{2}} (\int_{0}^{v_{1}} f (t_{1}, t_{2}) d t_{1}) d t_{2} = : J_{2} (v_{2})$ existent uniformément dans $0 < v_{1}, v_{2} < \infty$ , respectivement, et on a $\lim_{v_{1} \to \infty} J_{1} (v_{1}) = \lim_{v_{2} \to \infty} J_{2} (v_{2}) = \lim_{v_{1}, v_{2} \to \infty} \int_{0}^{v_{1}} \int_{0}^{v_{2}} f (t_{1}, t_{2}) d t_{1} d t_{2}$ . Ceci peut être considéré comme une généralisation du théorème de Fubini concernant lʼintégration successive au cas où $f \notin L^{1} ({\bar{R}}_{+}^{2})$ .

Let the function $f : {\bar{R}}_{+}^{m} \to C$ be such that $f \in L_{loc}^{1} ({\bar{R}}_{+}^{m})$ , where $m ⩾ 2$ is a fixed integer. We investigate the convergence behavior of the m-multiple integral

\int_{0}^{v_{1}} \int_{0}^{v_{2}} \dots \int_{0}^{v_{m}} f (t_{1}, t_{2}, \dots, t_{m}) d t_{1} d t_{2} \dots d t_{m} as \min {v_{1}, v_{2}, \dots, v_{m}} \to \infty,

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

f \in L_{loc}^{1} ({\bar{R}}_{+}^{2})

and the double integral (⁎) converges regularly, then the finite limits

\lim_{v_{2} \to \infty} \int_{0}^{v_{1}} (\int_{0}^{v_{2}} f (t_{1}, t_{2}) d t_{2}) d t_{1} = : J_{1} (v_{1})

and

\lim_{v_{1} \to \infty} \int_{0}^{v_{2}} (\int_{0}^{v_{1}} f (t_{1}, t_{2}) d t_{1}) d t_{2} = : J_{2} (v_{2})

exist uniformly in

0 < v_{1}, v_{2} < \infty

, respectively, and

\lim_{v_{1} \to \infty} J_{1} (v_{1}) = \lim_{v_{2} \to \infty} J_{2} (v_{2}) = \lim_{v_{1}, v_{2} \to \infty} \int_{0}^{v_{1}} \int_{0}^{v_{2}} f (t_{1}, t_{2}) d t_{1} d t_{2}

. This can be considered as a generalized version of Fubiniʼs theorem on successive integration in the case when

f \notin L^{1} ({\bar{R}}_{+}^{2})

.

Reçu le : 2012-01-26
Accepté le : 2012-03-07
Publié le : 2012-05-01

DOI : 10.1016/j.crma.2012.03.010

Affiliations des auteurs :

Ferenc Móricz ¹

¹ Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged 6720, Hungary

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     author = {Ferenc M\'oricz},
     title = {On the regular convergence of multiple integrals of locally {Lebesgue} integrable functions over $ {\overline{\mathbb{R}}}_{+}^{m}$},
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     pages = {459--464},
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     doi = {10.1016/j.crma.2012.03.010},
     language = {en},
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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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