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

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