Research ArticleOpen AccessGoogle Scholar indexed
Functions of Bounded (<i>p</i>(⋅), 2)-Variation in De la Vallée Poussin-Wiener’s Sense with Variable Exponent
Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
University of Barcelona, Spain
Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
- 1 Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
- 2 University of Barcelona, Spain
- 3 Departamento de Matemática, Universidad Central de Venezuela, Caracas, Venezuela
Advances in Pure Mathematics·Volume 07 (2017)·Pages 507–532·Published 23 August 2017·DOI10.4236/apm.2017.79033
Copy link · social · email
Abstract
In this paper we establish the notion of the space of bounded ( p (⋅), 2) variation in De la Vallée Poussin-Wiener’s sense with variable exponent. We show some properties of this space and we show that any uniformly bounded composition operator that maps this space into itself necessarily satisfies the so-called Matkowski’s conditions.
KeywordsGeneralized VariationDe la Vallée Poussin(<i>p</i>(&sdot)2)-Variation in Wiener’s SenseVariable ExponentComposition OperatorMatkowski’s Condition
- Jordan, C. (1881) Sur la série de Fourier. [On the Fourier Serie]. Comptes Rendus de l’Académie des Sciences, 92, 228-230.
- De la Vallée Poussin, C.J. (1908) Sur la convergence des formules d’interpolation entre ordennées equidistantes. [On the Convergence of Interpolation Formulas between Equidistant Orders]. Bulletin de la Classe des Sciences, Academie Royale de Belgique, 314-410.
- Wiener, N. (1924) The Quadratic Variation of a Function and Its Fourier Coefficients. Journal of Mathematical Physics, 3, 72-94. https://doi.org/10.1002/sapm19243272
- 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. https://doi.org/10.1214/aos/1176325354
- 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. https://doi.org/10.1007/978-1-4612-1880-7_13
- Dudley, R.M. and Norvaisa, R. (1999) Differentiability of Six Operators on Nonsmooth Functions and p-Variation. Lecture Notes in Math, 1703, Springer, Berlin. https://doi.org/10.1007/BFb0100744
- Appel, J., Banas, J. and Merentes, N. (2014) Bounded Variation and around. De Gruyter, Boston.
- Chistyakov, V.V. and Galkin, O.E. (1998) On Maps of Bounded p-Variation with . Positivity, 2, 19-45. https://doi.org/10.1023/A:1009700119505
- Diening, L. (2004) Maximal Function on Generalize Lebesgue Spaces L p(x) . Mathematical Inequalities & Applications, 7, 245-253. https://doi.org/10.7153/mia-07-27
- Azroul, E., Barbara, A. and Redwane, H. (2014) Existence and Nonexistence of a Solution for a Nonlinear p(x)-Elliptic Problem with Right-Hand Side Measure. International Journal of Analysis, 2014, 1-15.
- Fan, X., Zhao, Y. and Zhao, D. (2001) Compact Imbedding Theorems with Symmetry of Strauss-Lions Type for the Space W 1,p(x) Ω. Journal of Mathematical Analysis and Applications, 255, 333-348. https://doi.org/10.1006/jmaa.2000.7266
- Yin, L., Liang, Y., Zhang, Q. and Zhao, C. (2015) Existence of Solutions for a Variable Exponent System without PS Conditions. Journal of Differential Equations, 2015, 1-23.
- 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.