Vol. 12, No. 3, 2019

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 n , the Dirichlet problem in +n with boundary data in BMO(n1) is well-posed in the class of functions u for which the Littlewood–Paley measure associated with u, namely

dμu(x,t) := |u(x,t)|2tdxdt,

is a Carleson measure in +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(n1) can be characterized as the collection of nontangential pointwise traces of smooth null-solutions u to the elliptic system L with the property that μu is a Carleson measure in +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 +n of any given smooth null-solutions u of L in +n satisfying the above Carleson measure condition actually belongs to Sarason’s space VMO(n1) if and only if μu(T(Q))|Q| 0 as |Q| 0, uniformly with respect to the location of the cube Q n1 (where T(Q) 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 +n for a system L as above in the case when the boundary data are prescribed in Morrey–Campanato spaces in n1 . 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(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.

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
Milestones
Received: 22 March 2017
Revised: 29 April 2018
Accepted: 30 May 2018
Published: 7 October 2018
Authors
José María Martell
Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM
Consejo Superior de Investigaciones Científicas
Madrid
Spain
Department of Mathematics
University of Missouri
Columbia, MO
United States
Dorina Mitrea
Department of Mathematics
University of Missouri
Columbia, MO
United States
Irina Mitrea
Department of Mathematics
Temple University
Philadelphia, PA
United States
Marius Mitrea
Department of Mathematics
University of Missouri
Columbia, MO
United States