The Dirichlet problem for elliptic systems with data in K\"othe function spaces

TL;DR

This paper establishes the equivalence between the boundedness of the Hardy-Littlewood maximal operator on K"othe function spaces and the well-posedness of elliptic system Dirichlet problems with data in these spaces, extending classical results.

Contribution

It characterizes the well-posedness of elliptic systems' Dirichlet problems in various function spaces via maximal operator boundedness, including classical and weighted spaces.

Findings

Well-posedness of Dirichlet problems in Lebesgue, Lorentz, Zygmund, and Hardy spaces.

Uniqueness of the associated Poisson kernel for such systems.

Existence of a boundary trace for null-solutions via Fatou-type theorem.

Abstract

We show that the boundedness of the Hardy-Littlewood maximal operator on a K\"othe function space ${\mathbb{X}}$ and on its K\"othe 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 ${\rm HA}^p$ 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.