Norm-attaining Composition Operators on Lipschitz Spaces

Abstract

Every composition operator C ϕ on the Lipschitz space Lip 0 (X) attains its norm. This fact is essentially known and we give in this paper a sequential characterization of the extremal functions for the norm of C ϕ on Lip 0 (X). We also characterize the norm-attaining composition operators C ϕ on the little Lipschitz space lip 0 (X) which separates points uniformly and identify the extremal functions for the norm of C ϕ on lip 0 (X). We deduce that compact composition operators on lip 0 (X) are norm-attaining whenever the sphere unit of lip 0 (X) separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Hölder compact metric spaces (X, d α ) with 0 < α < 1.