02-6   Berichtsreihe des Mathematischen Seminars der Universität Kiel

## Non-Solvability for a class of left-invariant second-order differential operators on the Heisenberg group

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group $\bH_n$, of the form $${\cal P}_\Lambda= \sum_{i,j=1}^{n} \lambda_{ij}X_i Y_j={\,}^t X\Lambda Y,$$ where $\Lambda=(\lambda_{ij})$ is a complex $n\times n$ matrix. Such operators never satisfy a cone condition in the sense of Sj\"ostrand and H\"ormander (see e.g. \cite{MR2}). We may assume that ${\cal P}_\Lambda$ cannot be viewed as a differential operator on a lower dimensional Heisenberg group. Under the mild condition that $\Re\Lambda,$ $\Im\Lambda$ and their commutator are linearly independent, we show that ${\cal P}_\Lambda$ is not locally solvable, even in the presence of lower order terms, provided that $n\ge7$. In the case $n=3$ we show that there are some operators of the form described above that are locally solvable. This result extends to the Heisenberg group $\bH_3$ a phenomenon first observed in \cite{KM} in the case of $\bH_2.$ It is interesting to notice that the analysis of the exceptional operators for the case $n=3$ turns out to be more elementary than in the case $n=2.$ When $3\le n\le 6$ the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

Mathematics Subject Classification (1991): 35A05, 35D05, 43A80

Bibliographical note: wird in Trans. A.M.S. erscheinen

Keywords: Local solvability, Heisenberg group