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 $T_{f}[g]=f\circ g$, when $g$ belongs to a certain subspace $X$ of the space $BMO(\euclid)$ 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 $X$.