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

Abstract

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 $BMO\left({ℝ}^{n-1}\right)$ is well-posed in the class of functions $u$ for which the Littlewood–Paley measure associated with $u$, namely

$$d{\mu }_u\left(x^{\prime },t\right):=|\nabla u\left(x^{\prime },t\right){|}^{2}td{x}^{\prime }dt,$$

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 $BMO\left({ℝ}^{n-1}\right)$ 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 $VMO\left({ℝ}^{n-1}\right)$ if and only if ${\mu }_u\left(T\left(Q\right)\right)∕\left|Q\right|\to 0$ as $\left|Q\right|\to 0$, uniformly with respect to the location of the cube $Q\subset {ℝ}^{n-1}$ (where $T\left(Q\right)$ is the Carleson box associated with $Q$, and $\left|Q\right|$ 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 ${ℝ}^{n-1}$. 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 Hölder 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 Calderón–Zygmund operator $T$ satisfying $T\left(1\right)=0$ 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.