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 Tf defined by Tf [g] = f ◦ g, when g belongs to a certain subspace X of the space BMO(Rn) 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 Tf maps X to itself, those f ’s for which Tf is continuous from X to itself, and those f ’s for which Tf is differentiable in X.