Functional calculus on BMO and related spaces

Let f be a Borel measurable function of the complex plane to itself. We consider the nonlinear operator T-f defined by T-f [g] = f o g. when g belongs to a certain subspace X of the space BMO(R-n) of functions with bounded mean oscillation on the Euclidean space. In particular, we investigate the case in which X is the whole of BMO. the case in which X is the space VMO of functions with vanishing mean oscillation, and the case in which X is the closure in BMO of the smooth functions with compact support, We characterize those f's for which T-f maps X to itself, those f's for which T-f is continuous from X to itself, and those f's for which T-f is differentiable in X. (C) 2002 Elsevier Science (USA).