LOCAL SOLVABILITY OF 1ST-ORDER DIFFERENTIAL-OPERATORS NEAR A CRITICAL-POINT, OPERATORS WITH QUADRATIC SYMBOLS AND THE HEISENBERG-GROUP

We study questions of solvability for operators of the form p(x, D) + b, where p(x, xi) is a real quadratic form and b is-an-element-of C. As. one consequence, we obtain a necessary and sufficient condition for the local solvability of operators of the form L = [GRAPHICS] a(ij)x(j)partial derivative(i) + b, (a(ij) is-an-element-of R) near the critical point x = 0, and prove the existence of tempered fundamental solutions whenever L is locally solvable. Our analysis of these operators is largely based on recent results [MR II] about the solvability of left-invariant second order differential operators on the Heisenberg group and a transference principle for the Schrodinger representation.