Littlewood Subordination Theorem and Composition Operators on Function Spaces with Variable Exponents

Document Type : Research Paper


1 Department of Mathemathics, Faculty of Science, Payame Noor University (PNU), P. O. Box 19395-4697, Tehran , Iran.

2 Department of Pure Mathemathics, Faculty of Science, Imam Khomeini International University, Qazvin 34149, Iran.



This study concerns a detailed analysis of composition operators $C_\varphi$ on the classical Bergman spaces, as well as on the Hardy and Bergman spaces with variable exponents. Here, $\varphi$ is an analytic self-map of the open unit disk in the complex plane.
Accordingly, conditions for the boundedness of these operators are obtained. It is worth mentioning that the Littlewood subordination theorem plays a fundamental role in proving the stated theorems in which we use the Rubio de Francia extrapolation theorem.


[1] G.A. Chac\'{o}n and G.R. Chac\'{o}n, Analytic variable exponent Hardy spaces, Adv. Oper. Theory., 4 (2019), 738-749.
[2] G.R. Chac\'{o}n and H. Rafeiro, Variable exponent Bergman spaces, Nonlinear Analysis: Theory, Methods, and 
      Applications, 105} (2014), 41-49.
[3] G.R. Chac\'{o}n, H. Rafeiro and J.C. Vallejo, Carleson Measures on Variable exponent Bergman spaces, Complex Anal. 
     Oper. Theory, 11 (2017), 1623-1638.
[4] C.C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, Boca Raton, 1994.
[5] D. Cruz-Uribe and A. Fiorenza, Variable Exponent Lebesgue Spaces: Foundations and Harmonic Analysis, Birkhauser, 
     Basel, Switzerland, 2013.
[6] D. Cruz-Uribe, A. Fiorenza and C. Neugebauer, The maximal function on variable $L^p$ spaces, Ann. Acad. Sci. Fenn. 
      Math., 28 (2004), 223-238.
[7] L. Diening, Maximal function on generalized Lebesgue spaces $L^{p(\cdot)}$, Math. Inequal. Appl., 7 (2004), 245-253.
[8] L. Diening, P. H\"{a}st\"{o}, P. Harjulehto and M. R{\aa}u\v{z}i\v{c}ka, Lebesgue and Sobolev spaces with variable 
      exponents, Springer-Verlar, Berlin, 2011.
[9] P.L. Duren, Theory of Hp Spaces, Academic Press, New York, 1970.
[10] P. Duren and A. Schuster, Bergman Spaces}, Mathematical Surveys and Monographs, vol. 100, American 
       Mathematical Society, 2004.
[11] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Springer, New York, 2000.
[12] V. Kokilashvili and V. Paatashvili, On Hardy classes of analytic functions with a variable exponent, Proc. A. Razmadze
       Math. Inst., 142 (2006), 134-137.
[13] V. Kokilashvili and V. Paatashvili, On the convergence of sequences of functions in Hardy classes with a variable 
       exponent, Proc. A. Razmadze Math. Inst., 146 (2008), 124-126.
[14] O. Kov\'{a}\v{c}ik and J. R\'{a}konsn\'{\i}k, On spaces $L^{p(x)}$ and $W^{p(x)}$, Czechoslovak Math. J., 41(4) (1991), 
[15] J. Miao and D. Zheng, Compact operators on Bergman spaces, Integral Eq. Oper. Theory, 48(1) (2004), 61-79.
[16] H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen Co. Ltd., Tokyo, 1950.
[17] H. Nakano, Topology of Linear Topological Spaces, Maruzen Co. Ltd., Tokyo, 1951.
[18] A. Nekvinda, Hardy-Littlewood maximal operator on $L^{p(x)}(R^n)$, Math. Inequal. Appl., 7 (2004), 255-266.
[19] W. Orlicz, Uber konjugierte exponentenfolgen (German), Studia Math., 3 (1931), 200-211.
[20] W. Rudin, Real and Complex Analysis, Mac Graw-Hill, New York, 1970.
[21] I. Sharapudinov, On the topology of the space $L^{p(t)}([0; 1])$, Math. Notes, 26 (1979), 796-806.
[22] M. Stessin and K. Zhu, Composition operators induced by symbols defined on a polydisk, J. Math. Anal. Appl., 319 
       (2006), 815-829.
[23] M. Stessin and K. Zhu, Composition operators on embedded disks, J. Operator Theory, 56 (2006), 423-449.