We prove that for any homogeneous, second-order, constant complex coefficient elliptic system in , the Dirichlet problem in with boundary data in is well-posed in the class of functions for which the Littlewood–Paley measure associated with , namely
is a Carleson measure in .
In the process we establish a Fatou-type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions of such systems satisfying the said Carleson measure condition. In concert, these results imply that the space can be characterized as the collection of nontangential pointwise traces of smooth null-solutions to the elliptic system with the property that is a Carleson measure in .
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 of any given smooth null-solutions of in satisfying the above Carleson measure condition actually belongs to Sarason’s space if and only if as , uniformly with respect to the location of the cube (where is the Carleson box associated with , and denotes the Euclidean volume of ).
Moreover, we are able to establish the well-posedness of the Dirichlet problem in for a system as above in the case when the boundary data are prescribed in Morrey–Campanato spaces in . In such a scenario, the solution 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 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 Hölder or Lipschitz condition). This improves on Sarason’s classical result describing 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 Calderón–Zygmund operator satisfying extends as a linear and bounded mapping from (modulo constants) into itself. In turn, this is used to describe algebras of singular integral operators on , and to characterize the membership to via the action of various classes of singular integral operators.
"The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO." Anal. PDE 12 (3) 605 - 720, 2019. https://doi.org/10.2140/apde.2019.12.605