  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.
