Norm-attaining composition operators on Lipschitz spaces

Abstract

Every composition operator C_{\varphi} 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_{\varphi} on Lip_0(X). We also characterize the norm-attaining composition operators C_{\varphi} on the little Lipschitz space lip_0(X) which separates points uniformly and identify the extremal functions for the norm of C_{\varphi} 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 satisfi ed by spaces of little Lipschitz functions on Hölder compact metric spaces (X,d^{\alpha}) with 0<\alpha<1.