  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.
