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
scarora@maths.du.ac.in
Department of Mathematics PGDAV College
University of Delhi
Delhi-110065 India
gopaldatt@maths.du.ac.in
Department of Mathematics SGTB Khalsa College
University of Delhi
Delhi-110007 India
vermas@maths.du.ac.in
Received 29 April 2005; revised 22 July 2006
Communicated by A. J. Pryde

Abstract

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.

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2000 Mathematics Subject Classification: primary 47B33, 47B38; secondary 46E30, 46E35
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