APM  Vol.2 No.1 , January 2012
On Bounded Second Variation
Abstract: In this paper, we discuss various aspects of the problem of space-invariance, under compositions, of certain subclasses of the space of all continuously differentiable functions on an interval [a,b] We present a result about integrability of products of the form gοf.f'f(k)under suitable mild conditions and, finally, we prove that a Nemytskij operator Sg maps BV''[a,b] a distinguished subspace of the space of all functions of second bounded variation, into itself if, and only if, g BV''loc(R) A similar result is obtained for the space of all functions of bounded (p,2)-variation (1≤p≤1), A2p
Cite this paper: J. Giménez, L. López, N. Merentes and J. Sánchez, "On Bounded Second Variation," Advances in Pure Mathematics, Vol. 2 No. 1, 2012, pp. 22-26. doi: 10.4236/apm.2012.21005.

[1]   C. Jordan, “Sur la Serie de Fourier,” Comptes Rendus des Séances de l’Académie des Sciences, Vol. 2, 1881, pp. 228-230.

[2]   R. Kannan and C. K. Krueger, “Advanced Analysis on the Real Line,” Springer, New York, 1996.

[3]   V. I. Burenkov, “On Integration by Parts and a Problem on Composition of Absolutely Continuous Functions Which Arises in This Connection, Theory of Functions and Its Applications,” Proceedings of the Steklov Institute of Mathematics, Vol. 134, 1975, pp. 38-46.

[4]   J. Appell and P. P. Zabrejko, “Nonlinear Superposition Operator,” Cambridge University Press, New York, 1990.

[5]   J. Appell, Z. Jesús and O. Mejía, “Some Remarks on Nonlinear Composition Operators in Spaces of Differen- tiable Functions,” Bollettino della Unione Matematica Italiana—Serie IX, Vol. 4, No. 3, 2011, pp. 321-336.

[6]   M. Josephy, “Composing Functions of Bounded Varia- tion,” Proceedings of the AMS—American Mathematical Society, Vol. 83, No. 2, 1981, pp. 354-356. doi:10.1090/S0002-9939-1981-0624930-9

[7]   Ch. J. de la Valle Poussin, “Sur L’ntegrale de Lebesgue,” Transactions of the AMS—American Mathematical Society, Vol. 16, 1915, pp. 435-501.

[8]   F. Riesz, “Sur Certains Systmes Singuliers d’Quations Intgrales, Annales de l’Ecole Normale,” Suprieure, Vol. 28, No. 3, 1911, pp. 33-62.

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

[10]   J. Appell, “Some Counterexamples for Your Calculus Course,” Analysis, Vol. 31, No. 1, 2010, pp. 1001-1012.

[11]   G. Leoni, “A First Course in Sobolev Spaces,” American Mathematical Society, Vol. 105, Rode Island, 2009.

[12]   J. Appell and P. P. Zabrejko, “Remarks on the Superposi- tion Operator Problem in Various Function Spaces,” Complex Variables and Elliptic Equations, Vol. 55, No. 8, 2010, pp. 727-737. doi:10.1080/17476930903568332

[13]   N. Merentes, “On the Composition Operator in AC [a,b],” Collectanea Mathematica, Vol. 42, No. 1, 1991, pp. 121- 127.

[14]   N. Merentes and S. Rivas, “El Operador de Composicin en Espacios de Funciones con algún tipo de Variación Acotada,” IX Escuela Venezolana de Matemáticas, Facultad de Ciencias-ULA, Mérida, 1996.