99-13   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Local solvability for a class of second order complex coefficient differential operators on the Heisenberg group H2.

Let L be a differential operator on R5 whose principal part is of the form $\sum_{j,k=1}^2 \al_{jk} X_j Y_k$, where $X_j=\partial_{x_j}$ and $Y_j=\partial_{y_j}+x_j\partial_u$ , $j=1,2$, are the usual vector fields generating the Lie algebra of the Heisenberg group H2. We study the problem of local solvability of these doubly characteristic operators. The whole class of operators splits into three subclasses, depending on the sign of a respective determinant. The operators in the first subclass, when the determinant is negative, are generically non-solvable. The operators in the second subclass, when the determinant is positive, are solvable, for arbitrary left-invariant lower order terms, provided that the coefficient matrix $(\al_{j,k})$ is non-degenerate. This fact seems remarkable, since many of these operators have the property that the values taken by their principal symbol are not contained in any proper subcone of the complex plane. Under suitable conditions, solvability even holds in the presence of non-invariant lower order terms.

Mathematics Subject Classification (1991): 22E30, 35A07,35D05,43A80

Bibliographical note: erschienen in Math. Ann. 320 (2001), 731-755