Let be a metric space and denote the space of all bounded Lipschitz functions on the metric space . A mapping is called supercontractive if, for , there exists such that whenever . The authors prove that , the composition operator defined by , is compact on if and only if is supercontractive and is totally bounded in .
This result improves upon previous work of H. Kamowitz and S. Scheinberg [Stud. Math. 96, No. 3, 255–261 (1990; Zbl 0713.47030)] that studied the case of compact metric spaces.