## 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

#### 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 $BMO ( ℝ n − 1 )$ 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 $ℝ + n$.

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 $BMO ( ℝ n − 1 )$ 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 $ℝ + n$.

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 $VMO ( ℝ n − 1 )$ if and only if $μ u ( T ( Q ) ) ∕ | Q | → 0$ as $| Q | → 0$, uniformly with respect to the location of the cube $Q ⊂ ℝ n − 1$ (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 $ℝ + n$ 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 $T ( 1 ) = 0$ 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

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

Dates
Revised: 29 April 2018
Accepted: 30 May 2018
First available in Project Euclid: 25 October 2018

https://projecteuclid.org/euclid.apde/1540432867

Digital Object Identifier
doi:10.2140/apde.2019.12.605

Mathematical Reviews number (MathSciNet)
MR3864207

Zentralblatt MATH identifier
1298.35071

#### Citation

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. https://projecteuclid.org/euclid.apde/1540432867

#### References

• S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I”, Comm. Pure Appl. Math. 12 (1959), 623–727.
• S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, II”, Comm. Pure Appl. Math. 17 (1964), 35–92.
• R. Alvarado and M. Mitrea, Hardy spaces on Ahlfors-regular quasi-metric spaces: a sharp theory, Lecture Notes in Math. 2142, Springer, 2015.
• S. Axler, P. Bourdon, and W. Ramey, Harmonic function theory, 2nd ed., Graduate Texts in Math. 137, Springer, 2001.
• G. Bourdaud, “Remarques sur certains sous-espaces de ${\rm BMO}(\mathbb{R}^n)$ et de ${\rm bmo}(\mathbb{R}^n)$”, Ann. Inst. Fourier $($Grenoble$)$ 52:4 (2002), 1187–1218.
• M. Christ, Lectures on singular integral operators, CBMS Regional Conference Series in Math. 77, Amer. Math. Soc., Providence, RI, 1990.
• M. Christ and J.-L. Journé, “Polynomial growth estimates for multilinear singular integral operators”, Acta Math. 159:1-2 (1987), 51–80.
• R. R. Coifman, Y. Meyer, and E. M. Stein, “Some new function spaces and their applications to harmonic analysis”, J. Funct. Anal. 62:2 (1985), 304–335.
• B. E. J. Dahlberg and C. E. Kenig, “Hardy spaces and the Neumann problem in $L^p$ for Laplace's equation in Lipschitz domains”, Ann. of Math. $(2)$ 125:3 (1987), 437–465.
• M. Dindos, C. Kenig, and J. Pipher, “BMO solvability and the $A_\infty$ condition for elliptic operators”, J. Geom. Anal. 21:1 (2011), 78–95.
• X. T. Duong, L. Yan, and C. Zhang, “On characterization of Poisson integrals of Schrödinger operators with BMO traces”, J. Funct. Anal. 266:4 (2014), 2053–2085.
• E. B. Fabes and U. Neri, “Dirichlet problem in Lipschitz domains with BMO data”, Proc. Amer. Math. Soc. 78:1 (1980), 33–39.
• E. B. Fabes, R. L. Johnson, and U. Neri, “Spaces of harmonic functions representable by Poisson integrals of functions in BMO and $\mathscr{L}\sb{p,\lambda}$”, Indiana Univ. Math. J. 25:2 (1976), 159–170.
• C. Fefferman, “Characterizations of bounded mean oscillation”, Bull. Amer. Math. Soc. 77 (1971), 587–588.
• C. Fefferman and E. M. Stein, “$H\sp{p}$ spaces of several variables”, Acta Math. 129:3-4 (1972), 137–193.
• J. García-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Studies 116, North-Holland, Amsterdam, 1985.
• L. Grafakos, Classical and modern Fourier analysis, Pearson, Upper Saddle River, NJ, 2004.
• L. L. Helms, Introduction to potential theory, Pure and Applied Math. 22, Wiley, New York, 1969.
• S. Hofmann, M. Mitrea, and M. E. Taylor, “Symbol calculus for operators of layer potential type on Lipschitz surfaces with VMO normals, and related pseudodifferential operator calculus”, Anal. PDE 8:1 (2015), 115–181.
• S. Hofmann, D. Mitrea, M. Mitrea, and A. J. Morris, $L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc. 1159, Amer. Math. Soc., Providence, RI, 2017.
• T. Iwaniec and G. Martin, Geometric function theory and non-linear analysis, Oxford Univ. Press, 2001.
• J.-L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, I, Grundlehren der Math. Wissenschaften 181, Springer, 1972.
• J. M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea, “The higher order regularity Dirichlet problem for elliptic systems in the upper-half space”, pp. 123–141 in Harmonic analysis and partial differential equations, edited by P. Cifuentes et al., Contemp. Math. 612, Amer. Math. Soc., Providence, RI, 2014.
• J. M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea, “The Dirichlet problem for elliptic systems with data in Köthe function spaces”, Rev. Mat. Iberoam. 32:3 (2016), 913–970.
• J. M. Martell, D. Mitrea, I. Mitrea, and M. Mitrea, “On the $L^p$-Poisson semigroup associated with elliptic systems”, Potential Anal. 47:4 (2017), 401–445.
• V. G. Maz'ya and T. O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Monographs and Studies in Math. 23, Pitman, Boston, 1985.
• V. Maz'ya, M. Mitrea, and T. Shaposhnikova, “The Dirichlet problem in Lipschitz domains for higher order elliptic systems with rough coefficients”, J. Anal. Math. 110:1 (2010), 167–239.
• Y. Meyer, “Real analysis and operator theory”, pp. 219–235 in Pseudodifferential operators and applications (Notre Dame, IN, 1984), edited by F. Trèves, Proc. Sympos. Pure Math. 43, Amer. Math. Soc., Providence, RI, 1985.
• Y. Meyer, Ondelettes et opérateurs, II: Opérateurs de Calderón–Zygmund, Hermann, Paris, 1990.
• N. G. Meyers, “Mean oscillation over cubes and Hölder continuity”, Proc. Amer. Math. Soc. 15:5 (1964), 717–721.
• M. Mitrea, Clifford wavelets, singular integrals, and Hardy spaces, Lecture Notes in Math. 1575, Springer, 1994.
• D. Mitrea, Distributions, partial differential equations, and harmonic analysis, Springer, 2013.
• D. Mitrea, I. Mitrea, M. Mitrea, and M. Taylor, The Hodge–Laplacian: boundary value problems on Riemannian manifolds, de Gruyter Studies in Math. 64, de Gruyter, Berlin, 2016.
• D. Mitrea, M. Mitrea, and J. Verdera, “Characterizing regularity of domains via the Riesz transforms on their boundaries”, Anal. PDE 9:4 (2016), 955–1018.
• D. Mitrea, I. Mitrea, and M. Mitrea, “A sharp divergence theorem with non-tangential pointwise traces and applications”, submitted manuscript, 2017.
• J. Pipher and G. Verchota, “The Dirichlet problem in $L^p$ for the biharmonic equation on Lipschitz domains”, Amer. J. Math. 114:5 (1992), 923–972.
• W. Rudin, Real and complex analysis, 3rd ed., McGraw-Hill, New York, 1987.
• D. Sarason, “Functions of vanishing mean oscillation”, Trans. Amer. Math. Soc. 207 (1975), 391–405.
• D. Siegel, “The Dirichlet problem in a half-space and a new Phragmén–Lindelöf principle”, pp. 208–217 in Maximum principles and eigenvalue problems in partial differential equations (Knoxville, 1987), edited by P. W. Schaefer, Pitman Res. Notes Math. Ser. 175, Longman, Harlow, UK, 1988.
• D. Siegel and E. O. Talvila, “Uniqueness for the $n$-dimensional half space Dirichlet problem”, Pacific J. Math. 175:2 (1996), 571–587.
• V. A. Solonnikov, “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg, I”, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 665–706. In Russian.
• V. A. Solonnikov, “General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg, II”, Trudy Mat. Inst. Steklov. 92 (1966), 233–297. In Russian; translated in Proc. Steklov Inst. Math. 92 (1968), 269–339.
• E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Series 30, Princeton Univ. Press, 1970.
• E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Math. Series 43, Princeton Univ. Press, 1993.
• E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Series 32, Princeton Univ. Press, 1971.
• M. E. Taylor, Partial differential equations, I: Basic theory, 2nd ed., Applied Mathematical Sciences 115, Springer, 2011.
• M. E. Taylor, Partial differential equations, II: Qualitative studies of linear equations, 2nd ed., Applied Mathematical Sciences 116, Springer, 2011.
• M. E. Taylor, Partial differential equations, III: Nonlinear equations, 2nd ed., Applied Mathematical Sciences 117, Springer, 2011.
• A. Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Math. 123, Academic Press, Orlando, 1986.
• F. Wolf, “The Poisson integral: a study in the uniqueness of harmonic functions”, Acta Math. 74 (1941), 65–100.
• H. Yoshida, “A type of uniqueness for the Dirichlet problem on a half-space with continuous data”, Pacific J. Math. 172:2 (1996), 591–609.