The Dirichlet problem for elliptic systems with data in Köthe function spaces (Q346657)

Summary: 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.