  We prove that for any homogeneous, second order, constant complex coefficient
elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with
boundary data in BMO is well-posed in the class of functions $u$ with
$d\mu_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We
establish a Fatou type theorem guaranteeing the existence of the pointwise
nontangential boundary trace for smooth null-solutions $u$ of such systems
satisfying the said Carleson measure condition. These imply that BMO can be
characterized as the collection of nontangential pointwise traces of smooth
null-solutions $u$ to the elliptic system $L$ with the property that $\mu_u$ is
a Carleson measure.
  We establish a regularity result for the BMO-Dirichlet problem in the
upper-half space: the nontangential pointwise trace of any given smooth
null-solutions of $L$ satisfying the above Carleson measure condition belongs
to Sarason's space VMO if and only if $\mu_u$ satsifies a vanishing Carleson
measure condition. Moreover, we obtain the well-posedness of the Dirichlet
problems when the boundary data are prescribed in Morrey-Campanato and
solutions are required to satisfy a vanishing Carleson measure condition of
fractional order.
  As a consequence, we characterize the space VMO as the closure in BMO of
classes of smooth functions contained in BMO within which uniform continuity
may be suitably quantified (such as the class of smooth functions satisfying a
H\"older or Lipschitz condition), improving Sarason's classical result
describing VMO as the closure in BMO of the space of uniformly continuous
functions with bounded mean oscillations.
  Finally, we show that any Calder\'on-Zygmund operator $T$ satisfying $T(1)=0$
extends as a bounded linear operator in $\mathrm{VMO}$, and characterize the
membership to $\mathrm{VMO}$ via the action of various classes of singular
integral operators.
