# Revista Matemática Iberoamericana

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**Volume 32, Issue 3, 2016, pp. 913–970**

**DOI: 10.4171/RMI/903**

Published online: 2016-10-03

The Dirichlet problem for elliptic systems with data in Köthe function spaces

José María Martell^{[1]}, Dorina Mitrea

^{[2]}, Irina Mitrea

^{[3]}and Marius Mitrea

^{[4]}(1) Universidad Autónoma de Madrid, Spain

(2) University of Missouri, Columbia, United States

(3) Temple University, Philadelphia, USA

(4) University of Missouri, Columbia, USA

We show that the boundedness of the Hardy–Littlewood maximal operator on a Köthe function space $\mathbb X$ and on its Köthe dual $\mathbb X$' is equivalent to the well-posedness of the $\mathbb X$-Dirichlet and $\mathbb X$'-Dirichlet problems in$\mathbb R^n_+$ in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space $H^1$, and the Beurling-Hardy space HA$^p$ for $p \in (1,\infty)$. Based on the well-posedness of the $L^p$-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.

*Keywords: *Dirichlet problem, second-order elliptic system, nontangential maximal function, Hardy–Littlewood maximal operator, Poisson kernel, Green function, Köthe function space, Muckenhoupt weight, Lebesgue space, variable exponent Lebesgue space, Lorentz space, Zygmund space, Orlicz space, Hardy space, Beurling algebra, Hardy–Beurling space, semigroup, Fatou type theorem

Martell José María, Mitrea Dorina, Mitrea Irina, Mitrea Marius: The Dirichlet problem for elliptic systems with data in Köthe function spaces. *Rev. Mat. Iberoam.* 32 (2016), 913-970. doi: 10.4171/RMI/903