J. Aust. Math. Soc. 83 (2007), no. 3, pp. 327–334.

Weighted composition operators on Orlicz–Sobolev spaces

Subhash C. Arora Gopal Datt Satish Verma
Department of Mathematics University of Delhi
Delhi-110007 India
Department of Mathematics PGDAV College
University of Delhi
Delhi-110065 India
Department of Mathematics SGTB Khalsa College
University of Delhi
Delhi-110007 India
Received 29 April 2005; revised 22 July 2006
Communicated by A. J. Pryde


For an open subset \Omega of the Euclidean space \mathbf {R}^n, a measurable non-singular transformation T:\Omega \to \Omega and a real-valued measurable function u on \mathbf {R}^n, we study the weighted composition operator uC_T : f \mapsto u \cdot (f \circ T) on the Orlicz–Sobolev space W^{1, \varphi }(\Omega ) consisting of those functions of the Orlicz space L^{\varphi }(\Omega ) whose distributional derivatives of the first order belong to L^{\varphi }(\Omega ). We also discuss a sufficient condition under which uC_T is compact.

Download the article in PDF format (size 96 Kb)

2000 Mathematics Subject Classification: primary 47B33, 47B38; secondary 46E30, 46E35
(Metadata: XML, RSS, BibTeX)
indicates author for correspondence


  1. Robert A. Adams, Sobolev Spaces (Academic Press, New York, 1975). MR450957
  2. S. C. Arora and M. Mukherjee, ‘Compact composition operators on Sobolev spaces’, Indian J. Math. 37 (1995), 207–219. MR1458444
  3. Y. Cui, H. Hudzik, Romesh Kumar and L. Maligranda, ‘Composition operators in Orlicz spaces’, J. Aust. Math. Soc. 76 (2004), 189–206. MR2041244
  4. Herbert Kamowitz and Dennis Wortman, ‘Compact weighted composition operators on Sobolev related spaces’, Rocky Mountain J. Math. 17 (1987), 767–782. MR923745
  5. B. S. Komal and Shally Gupta, ‘Composition operators on Orlicz spaces’, Indian J. Pure Appl. Math. 32 (2001), 1117–1122. MR1846115
  6. A. Kufner, O. John and S. Fucik, Function Spaces (Noordhoff International Publishing, Leyden, 1977). MR482102
  7. Romesh Kumar, ‘Composition operators on Orlicz spaces’, Integral Equations Operator Theory 29 (1997), 17–22. MR1466857
  8. Richard L. Wheeden and Antoni Zygmund, Measure and Integral (Marcel Dekker Inc., New York, 1977). MR492146
Australian Mathematical Publishing Association Inc.

Valid XHTML 1.0 Transitional