The BMO-Dirichlet problem for elliptic systems in the upper-half space and quantitative characterizations of VMO

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.