SOLVABILITY OF INVARIANT DIFFERENTIAL-OPERATORS ON METABELIAN-GROUPS

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. © 1990 by Pacific Journal of Mathematics.