  In this paper we continue the study of important Banach spaces of slice
hyperholomorphic functions on the quaternionic unit ball by investigating the
BMO- and VMO-spaces of slice hyperholomorphic functions. We discuss in
particular conformal invariance and a refined characterization of these spaces
in terms of Carleson measures. Finally we show the relations with the Bloch and
Dirichlet space and the duality relation with the Hardy space
$\mathcal{H}^1(\mathbb{D})$.
  The importance of these spaces in the classical theory is well known. It is
therefore worthwhile to study their slice hyperholomorphic counterparts, in
particular because slice hyperholomorphic functions were found to have several
applications in operator theory and Schur analysis.
