In this work we use
non-commutative harmonic analysis in the study of differential operators on a certain
class of solvable Lie groups. A left invariant differential (a differential operator that
commutes with left translations on the group) can be synthesized in terms of
differential operators on lower dimensional spaces. This synthesis is easily described
for a certain class of simply connected solvable Lie groups, those arising as
semi-direct products of simply connected abelian groups.
We derive sufficient conditions for the semiglobal solvability of left invariant
differential operators on such groups in terms of the lower dimensional differential
operators. These conditions are seen to be satisfied for certain classes of second
order differential operators, thus yielding semiglobal solvability. Specifically
elliptic, sub-elliptic, transversally elliptic and parabolic operators are
investigated.
