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 in R-n, the Dirichlet problem in R-+(n) with boundary data in BMO(Rn-1) is well-posed in the class of functions u for which the Littlewood-Paley measure associated with u, namely d mu(u)(x', t) := vertical bar del u(x', t)vertical bar(2) t dx' dt, is a Carleson measure in R-+(n). In the process 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. In concert, these results imply that the space BMO(Rn-1) 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 in R-+(n). We also establish a regularity result for the BMO-Dirichlet problem in the upper half-space, to the effect that the nontangential pointwise trace on the boundary of R-+(n) of any given smooth null-solutions u of L in R-+(n) satisfying the above Carleson measure condition actually belongs to Sarason's space VMO(Rn-1) if and only if mu(u)(T(Q))/vertical bar Q vertical bar -> 0 as vertical bar Q vertical bar -> 0, uniformly with respect to the location of the cube Q subset of Rn-1 (where T(Q) is the Carleson box associated with Q, and vertical bar Q vertical bar denotes the Euclidean volume of Q). Moreover, we are able to establish the well-posedness of the Dirichlet problem in R-+(n) for a system L as above in the case when the boundary data are prescribed in Morrey-Campanato spaces in Rn-1. In such a scenario, the solution u is required to satisfy a vanishing Carleson measure condition of fractional order. By relying on these well-posedness and regularity results we succeed in producing characterizations of 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 Holder or Lipschitz condition). This improves on Sarason's classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. In turn, this allows us to show that any Calderon-Zygmund operator T satisfying T(1) = 0 extends as a linear and bounded mapping from VMO (modulo constants) into itself. In turn, this is used to describe algebras of singular integral operators on VMO, and to characterize the membership to VMO via the action of various classes of singular integral operators.