Norm-attaining Composition Operators on Lipschitz Spaces

Abstract

Every composition operator $C_{\varphi}$ on the Lipschitz space $\operatorname{Lip}_0(X)$ 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 $\operatorname{Lip}_0(X)$. We also characterize the norm-attaining composition operators $C_{\varphi}$ on the little Lipschitz space $\operatorname{lip}_0(X)$ which separates points uniformly and identify the extremal functions for the norm of $C_{\varphi}$ on $\operatorname{lip}_0(X)$. We deduce that compact composition operators on $\operatorname{lip}_0(X)$ are norm-attaining whenever the sphere unit of $\operatorname{lip}_0(X)$ separates points uniformly. In particular, this condition is satisfied by spaces of little Lipschitz functions on Hölder compact metric spaces $(X,d^{\alpha})$ with $0 \lt \alpha \lt 1$.