  We investigate a connection between solvability of the Dirichlet problem for
an infinitely degenerate elliptic operator and the validity of an
Orlicz-Sobolev inequality in the associated subunit metric space. For
subelliptic operators it is known that the classical Sobolev inequality is
sufficient and almost necessary for the Dirichlet problem to be solvable with a
quantitative bound on the solution [11]. When the degeneracy is of infinite
type, a weaker Orlicz-Sobolev inequality seems to be the right substitute [7].
In this paper we investigate this connection further and reduce the gap between
necessary and sufficient conditions for solvability of the Dirichlet problem.
