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The Space of Bounded p(·)-Variation in Wiener’s Sense with Variable Exponent

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DOI: 10.4236/apm.2015.511064    3,104 Downloads   3,534 Views   Citations

ABSTRACT

In this paper, we proof some properties of the space of bounded p(·)-variation in Wiener’s sense. We show that a functions is of bounded p(·)-variation in Wiener’s sense with variable exponent if and only if it is the composition of a bounded nondecreasing functions and hölderian maps of the variable exponent. We show that the composition operator H, associated with , maps the spaces into itself if and only if h is locally Lipschitz. Also, we prove that if the composition operator generated by maps this space into itself and is uniformly bounded, then the regularization of h is affine in the second variable.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Mejía, O. , Merentes, N. and Sánchez, J. (2015) The Space of Bounded p(·)-Variation in Wiener’s Sense with Variable Exponent. Advances in Pure Mathematics, 5, 703-716. doi: 10.4236/apm.2015.511064.

References

[1] Jordan, C. (1881) Sur la série de Fourier. Comptes Rendus de l’Académie des Sciences, Paris, 92, 228-230.
[2] Wiener, N. (1924) The Quadratic Variation of a Function and Its Fourier Coefficients. Journal of Mathematical Physics, 3, 73-94.
[3] Young, L.C. (1937) Sur une généralisation de la notion de variation de pussance piéme bornée au sens de M. Wiener, et sur la convergence des séries de Fourier. Comptes Rendus de l’Académie des Sciences, Paris, 204, 470-472.
[4] Love, E.R. and Young, L.C. (1937) Sur une classe de fonctionelles linéaires. Fundamenta Mathematicae, 28, 243-257.
[5] Dudley, R.M. (1994) The Order of the Remainder in Derivatives of Composition and Inverse Operators for p-Variation Norms. Annals of Statistics, 22, 1-20.
http://dx.doi.org/10.1214/aos/1176325354
[6] Dudley, R.M. (1997) Empirical Processes and p-Variation. In: Pollard, D., Torgersen, E. and Yang, G.L., Eds., Festschrift for Lucien Le Cam, Springer, New York, 219-233.
http://dx.doi.org/10.1007/978-1-4612-1880-7_13
[7] Dudley, R.M. and Norvaisa, R. (1999) Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Mathematics, 1703. Springer, Berlin.
[8] Appell, J., Banas, J. and Merentes, N. (2014) Bounded Variation and Around. De Gruyter, Boston.
[9] Chistyakov, V.V. and Galkin, O.E. (1998) On Maps of Bounded p-Variation with . Positivity, 2, 19-45.
http://dx.doi.org/10.1023/A:1009700119505
[10] Diening, L. (2004) Maximal Function on Generalized Lebesgue Spaces . Mathematical Inequalities & Applications, 7, 245-253.
http://dx.doi.org/10.7153/mia-07-27
[11] Azroul, E., Barbara, A. and Redwane, H. (2014) Existence and Nonexistence of a Solution for a Nonlinear -Elliptic Problem with Right-Hand Side Measure. International Journal of Analysis, 2014, Article ID: 320527.
http://dx.doi.org/10.1155/2014/320527
[12] Fan, X.L., Zhao, Y.Z. and Zhao, D. (2001) Compact Imbedding Theorems with Symmetry of Strauss-Lions Type for the Space . Journal of Mathematical Analysis and Applications, 255, 333-348.
http://dx.doi.org/10.1006/jmaa.2000.7266
[13] Yin, L., Liang, Y., Zhang, Q. and Zhao, C.S. (2015) Existence of Solutions for a Variable Exponent System without PS Conditions. Journal of Differential Equations, 2015, 1-23.
[14] Radulescu, V.D. and Repovs, D.D. (2015) Partial Differential Equations with Variable Exponent: Variational Methods and Qualitative Analysis. CRC Press, Taylor & Francis Group, Boca Raton.
[15] Orlicz, W. (1931) über konjugierte exponentenfolgen. Studia Mathematica, 3, 200-211.
[16] Nakano, H. (1950) Modulared Semi-Ordered Linear Spaces. Maruzen Co., Ltd., Tokyo.
[17] Nakano, H. (1951) Topology and Topological Linear Spaces. Maruzen Co., Ltd., Tokyo.
[18] Musielak, J. (1983) Orlicz Spaces and Modular Spaces. Lecture Notes in Mathematics. Vol. 1034, Springer-Verlag, Berlin.
[19] Musielak, J. and Orlicz, W. (1959) On Modular Spaces. Studia Mathematica, 18, 49-65.
[20] Kovácik, O. and Rákosník, J. (1991) On Spaces and . Czechoslovak Mathematical Journal, 41, 592-618.
[21] Castillo, R., Merentes, N. and Rafeiro, H. (2014) Bounded Variation Spaces with p-Variable. Mediterranean Journal of Mathematics, 11, 1069-1079.
http://dx.doi.org/10.1007/s00009-013-0342-5
[22] Federer, H. (1969) Geometric Measure Theory. Springer-Verlag, Heidelberg.
[23] Sierpiński, W. (1933) Sur une propriété des fonctions qui n’ont que des discontinuités de première espèce. Bulletin de la Section Scientifique de l’Académie Roumaine, 16, 1-4.
[24] Merentes, N. and Rivas, S. (1996) El Operador de Composición en Espacios de Funciones con Algún Tipo de Variación Acotada. IX Escuela Venezolana de Matemáticas, Facultad de Ciencias-ULA, Mérida.
[25] Appell, J. and Zabreiko, P.P. (1990) Nonlinear Superposition Operators. Cambridge University Press, Cambrige.
http://dx.doi.org/10.1017/CBO9780511897450
[26] Kuczma, M. (1885) An Introduction to the Theory of Functional Equations and Inequalities. Polish Scientific Editors and Silesian University, Warszawa-Krakow-Katowice.
[27] Matkowski, J. (2011) Uniformly Bounded Composition Operators between General Lipschitz Function Normed Spaces. Topological Methods in Nonlinear Analysis, 38, 395-405.

  
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