José María Martell, Dorina Mitrea, Irina Mitrea, Marius Mitrea

Anal. PDE 12 (3), 605-720, (2019) DOI: 10.2140/apde.2019.12.605
KEYWORDS: BMO Dirichlet problem, VMO Dirichlet problem, Carleson measure, vanishing Carleson measure, second-order elliptic system, Poisson kernel, Lamé system, nontangential pointwise trace, Fatou-type theorem, Hardy space, Holder space, Morrey–Campanato space, square function, quantitative characterization of VMO, dense subspaces of VMO, boundedness of Calderón–Zygmund operators on VMO, 35B65, 35C15, 35J47, 35J57, 35J67, 42B37, 35E99, 42B20, 42B30, 42B35

We prove that for any homogeneous, second-order, constant complex coefficient elliptic system $L$ in ${\mathbb{R}}^{n}$, the Dirichlet problem in ${\mathbb{R}}_{+}^{n}$ with boundary data in $BMO\left({\mathbb{R}}^{n-1}\right)$ 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 ${\mathbb{R}}_{+}^{n}$.

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 $BMO\left({\mathbb{R}}^{n-1}\right)$ 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 ${\mathbb{R}}_{+}^{n}$.

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 ${\mathbb{R}}_{+}^{n}$ of any given smooth null-solutions $u$ of $L$ in ${\mathbb{R}}_{+}^{n}$ satisfying the above Carleson measure condition actually belongs to Sarason’s space $VMO\left({\mathbb{R}}^{n-1}\right)$ if and only if ${\mu}_{u}\left(T\left(Q\right)\right)\u2215\left|Q\right|\to 0$ as $\left|Q\right|\to 0$, uniformly with respect to the location of the cube $Q\subset {\mathbb{R}}^{n-1}$ (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 ${\mathbb{R}}_{+}^{n}$ for a system $L$ as above in the case when the boundary data are prescribed in Morrey–Campanato spaces in ${\mathbb{R}}^{n-1}$. 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.