Analysis & PDE

  • Anal. PDE
  • Volume 12, Number 3 (2019), 605-720.

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

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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 ( n 1 ) is well-posed in the class of functions u for which the Littlewood–Paley measure associated with u , namely

d μ u ( x , t ) : = | u ( x , t ) | 2 t d x d t ,

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 ( n 1 ) can be characterized as the collection of nontangential pointwise traces of smooth null-solutions u to the elliptic system L with the property that μ u 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 ( n 1 ) if and only if μ u ( T ( Q ) ) | Q | 0 as | Q | 0 , uniformly with respect to the location of the cube  Q n 1 (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 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 ( 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.

Article information

Anal. PDE, Volume 12, Number 3 (2019), 605-720.

Received: 22 March 2017
Revised: 29 April 2018
Accepted: 30 May 2018
First available in Project Euclid: 25 October 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B65: Smoothness and regularity of solutions 35C15: Integral representations of solutions 35J47: Second-order elliptic systems 35J57: Boundary value problems for second-order elliptic systems 35J67: Boundary values of solutions to elliptic equations 42B37: Harmonic analysis and PDE [See also 35-XX]
Secondary: 35E99: None of the above, but in this section 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30: $H^p$-spaces 42B35: Function spaces arising in harmonic analysis

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


Martell, José María; Mitrea, Dorina; Mitrea, Irina; Mitrea, Marius. The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO. Anal. PDE 12 (2019), no. 3, 605--720. doi:10.2140/apde.2019.12.605.

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