Functional Calculus on BMO-type Spaces of Bourgain, Brezis and Mironescu

A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\mathbb{C}\to \mathbb{C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb{R}}^n$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.