UNIFORMLY BOUNDED COMPOSITION OPERATORS ON A BANACH SPACE OF BOUNDED WIENER-YOUNG VARIATION FUNCTIONS

Title & Authors
UNIFORMLY BOUNDED COMPOSITION OPERATORS ON A BANACH SPACE OF BOUNDED WIENER-YOUNG VARIATION FUNCTIONS
Glazowska, Dorota; Guerrero, Jose Atilio; Matkowski, Janusz; Merentes, Nelson;

Abstract
We prove, under some general assumptions, that a generator of any uniformly bounded Nemytskij operator, mapping a subset of space of functions of bounded variation in the sense of Wiener-Young into another space of this type, must be an affine function with respect to the second variable.
Keywords
$\small{{\varphi}}$-variation in the sense of Wiener;uniformly bounded operator;regularization;composition operator;Jensen equation;
Language
English
Cited by
1.
Uniformly Bounded Set-Valued Composition Operators in the Spaces of Functions of Bounded Variation in the Sense of Riesz, International Journal of Modern Nonlinear Theory and Application, 2015, 04, 04, 226
References
1.
J. Appell and P. P. Zabrejko, Nonlinear Superposition Operator, Cambridge University Press, New York, 1990.

2.
V. V. Chistyakov, Mappings of generalized variation and composition operators, J. Math. Sci. (New York) 110 (2002), no. 2, 2455-2466.

3.
J. A. Guerrero, H. Leiva, J. Matkowski, and N. Merentes, Uniformly continuous composition operators in the space of bounded ${\phi}$-variation functions, Nonlinear Anal. 72 (2010), no. 6, 3119-3123.

4.
M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Editors and Silesian University, Warszawa-Krakow-Katowice, 1985.

5.
K. Lichawski, J. Matkowski, and J. Mis, Locally defined operators in the space of differentiable functions, Bull. Polish Acad. Sci. Math. 37 (1989), no. 1-6, 315-325.

6.
W. A. Luxemburg, Banach Function Spaces, Ph.D. thesis, Technische Hogeschool te Delft, Netherlands, 1955.

7.
L. Maligranda and W. Orlicz, On some properties of functions of generalized variation, Monatsh. Math. 104 (1987), no. 1, 53-65.

8.
J. Matkowski, Functional equations and Nemytskii operators, Funkcial. Ekvac. 25 (1982), no. 2 127-132.

9.
J. Matkowski, Uniformly bounded composition operators between general Lipschitz function normed spaces, Topol. Methods Nonlinear Anal. 38 (2011), no. 2, 395-406.

10.
J. Matkowski, Uniformly continuous superposition operators in the spaces of bounded variation functions, Math. Nachr. 283 (2010), no. 7, 1060-1064.

11.
J. Matkowski and J. Mis, On a characterization of Lipschitzian operators of substitution in the space BV (a, b), Math. Nachr. 117 (1984), 155-159.

12.
J. Musielak and W. Orlicz, On generalized variations. I, Studia Math. 18 (1959), 11-41.

13.
H. Nakano, Modulared Semi-Ordered Linear Spaces, Tokyo, 1950.

14.
W. Orlicz, A note on modular spaces. I, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. 9 (1961), 157-162.

15.
N. Wiener, The quadratic variation of function and its Fourier coefficients, Massachusett J. Math. 3 (1924), 72-94.

16.
M. Wrobel, On functions of bounded n-th variation, Ann. Math. Sil. No. 15 (2001), 79-86.

17.
M. Wrobel, Lichawski-Matkowski-Mis theorem of locally defined operators for functions of several variables, Ann. Acad. Pedagog. Crac. Studia Math. 7 (2008), 15-22.

18.
M. Wrobel, Locally defined operators and a partial solution of a conjecture, Nonlinear Anal. 72 (2010), no. 1, 495-506.

19.
M. Wrobel, Representation theorem for local operators in the space of continuous and monotone functions, J. Math. Anal. Appl. 372 (2010), no. 1, 45-54.

20.
M. Wrobel, Locally defined operators in Holder's spaces, Nonlinear Anal. 74 (2011), no. 1, 317-323.

21.
L. C. Young, Sur une generalisation de la notion de variation de puissance p-ieme bornee au sens de N. Wiener, et sur la convergence des series de Fourier, C. R. Acad. Sci. 204 (1937), no. 7, 470-472.