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

Abstract

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.