APM  Vol.2 No.1 , January 2012
On Second Riesz Φ-Variation of Normed Space Valued Maps
ABSTRACT
In this article we present a Riesz-type generalization of the concept of second variation of normed space valued functions defined on an interval [a,b]R. We show that a function f [a,b], where X is a reflexive Banach space, is of bounded second Φ-variation, in the sense of Riesz, if and only if it can be expressed as the (Bochner) integral of a function of bounded (first) $\Phi$-variation. We provide also a Riesz lemma type inequality to estimate the total second Riesz-Φ-variation introduced.

Cite this paper
M. Bracamonte, J. Giménez, N. Merentes and J. Sánchez, "On Second Riesz Φ-Variation of Normed Space Valued Maps," Advances in Pure Mathematics, Vol. 2 No. 1, 2012, pp. 45-58. doi: 10.4236/apm.2012.21011.
References
[1]   Ch. J. de la Vallée Poussin, “Sur la Convergence des For- mules D’interpolation Entre Ordennées Equidistantes,” Bulletin de la Classe des Sciences, No. 4, 1908, pp. 314-410.

[2]   F. Riesz, “Sur Certains Systems Singuliers d’Equations Integrates,” Annales Scientifiques de l’école Normale Supérieure, Vol. 3, No. 28, 1911, pp. 33-68.

[3]   A. M. Russell and C. J. F. Upton, “A Generalization of a Theorem by F. Riesz,” Mathematical Analysis, Vol. 9, No. 1, 1983, pp. 69-77. doi:10.1007/BF01903991

[4]   N. Merentes, “On Functions of Bounded (p,2)-Variation,” Collectanea Mathematica, Vol. 43, No. 2, 1992, pp. 117- 123.

[5]   N. Merentes and S. Rivas, “El Operador de Composici? n en Espacios de Funciones con Algún Tipo de Variación Acotada,” IX Escuela Venezolana de Matemáticas, Facultad de Ciencias-ULA, Mérida, 1996.

[6]   A. W. Roberts and D. E. Varberg, “Functions of Bounded Convexity,” Bulletin of the AMS—American Mathematical Society, Vol. 75, No. 3, 1969, pp. 568-572. doi:10.1090/S0002-9904-1969-12244-5

[7]   F. Szigeti, “Composition of Sobolev Functions and Applications,” Notas de Matematicas, No. 86, 1987, pp. 1- 25.

[8]   L. Maligranda, “Orlicz Spaces and Interpolation,” Semi- nars in Mathematics 5, University of Campinas, Campi- nas, 1989.

[9]   J. Diestel and J. J. Uhl, “Vector Measures,” Mathematical Surveys, Vol. 15, 1977.

[10]   V. V. Chistyakov, “Mappings of Generalized Variation and Composition Operators,” Journal of Mathematical Sciences, Vol. 110, No. 2, 2002, pp. 2455-2466. doi:10.1023/A:1015018310969

[11]   V. Barbu and Th. Precupanu, “Convexity and Optimiza- tion on Banach Spaces,” Sijthof and Noordhoff, Nederlands, 1978.

 
 
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