Functional calculus on \(BMO\) and related spaces (Q1614764)

It is considered a nonlinear autonomous superposition operator \[ {\mathbf T}_f [g] = f\circ g \] on a certain subspace \(X\) of the space \(BMO({\mathbb R}^n)\). The question is to characterise those Borel measurable functions \(f\) for which the operator \({\mathbf T}_f\) is continuous or differentiable on \(X\). The following subspaces \(X\) are considered: \(BMO({\mathbb R}^n)\), \(VMO({\mathbb R}^n)\) and \(CMO({\mathbb R}^n)\) (closure in \(BMO\)-norm of \(C_c^{\infty}({\mathbb R}^n)\) as well as their respective inhomogeneous counterparts. Namely, \(X = bmo({\mathbb R}^n)\) (a linear subspace of those functions having finite mean on the unit cube in \({\mathbb R}^n\)), \(X = cmo({\mathbb R}^n)\) (closure in \(bmo\)-norm of \(C_c^{\infty}({\mathbb R}^n)\)), \(X = vmo({\mathbb R}^n) = VMO({\mathbb R}^n) \cap bmo({\mathbb R}^n)\). The results by Fominykh-Chevalier, Brezis-Nirenberg and Marcus-Mizel are extended.