Vol. 12, No. 3, 2019

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

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

Vol. 12 (2019), No. 3, 605–720
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

$d{\mu }_{u}\left({x}^{\prime },t\right):=|\nabla u\left({x}^{\prime },t\right){|}^{2}\phantom{\rule{0.3em}{0ex}}t\phantom{\rule{0.3em}{0ex}}d{x}^{\prime }\phantom{\rule{0.3em}{0ex}}dt,$

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 ${\mu }_{u}\left(T\left(Q\right)\right)∕|Q|\to 0$ as $|Q|\to 0$, 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 $|Q|$ 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.

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
Mathematical Subject Classification 2010
Primary: 35B65, 35C15, 35J47, 35J57, 35J67, 42B37
Secondary: 35E99, 42B20, 42B30, 42B35