  We study some basic analytic questions related to differential operators on
Lie manifolds, which are manifolds whose large scale geometry can be described
by a a Lie algebra of vector fields on a compactification. We extend to Lie
manifolds several classical results on Sobolev spaces, elliptic regularity, and
mapping properties of pseudodifferential operators. A tubular neighborhood
theorem for Lie submanifolds allows us also to extend to regular open subsets
of Lie manifolds the classical results on traces of functions in suitable
Sobolev spaces. Our main application is a regularity result on polyhedral
domains $\PP \subset \RR^3$ using the weighted Sobolev spaces $\Kond{m}a(\PP)$.
In particular, we show that there is no loss of $\Kond{m}a$--regularity for
solutions of strongly elliptic systems with smooth coefficients. For the proof,
we identify $\Kond{m}a(\PP)$ with the Sobolev spaces on $\PP$ associated to the
metric $r_{\PP}^{-2} g_E$, where $g_E$ is the Euclidean metric and $r_{\PP}(x)$
is a smoothing of the Euclidean distance from $x$ to the set of singular points
of $\PP$. A suitable compactification of the interior of $\PP$ then becomes a
regular open subset of a Lie manifold. We also obtain the well-posedness of a
non-standard boundary value problem on a smooth, bounded domain with boundary
$\maO \subset \RR^n$ using weighted Sobolev spaces, where the weight is the
distance to the boundary.
