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

Abstract:

We study the question of local solvability for second-order, left-invariant differential operators on the Heisenberg group $\mathbb {H}_n$, of the form \[ \mathcal {P}_\Lambda = \sum _{i,j=1}^{n} \lambda _{ij}X_i Y_j={ }^t X\Lambda Y, \] where $\Lambda =(\lambda _{ij})$ is a complex $n\times n$ matrix. Such operators never satisfy a cone condition in the sense of Sjöstrand and Hörmander. We may assume that $\mathcal {P}_\Lambda$ cannot be viewed as a differential operator on a lower-dimensional Heisenberg group. Under the mild condition that $\operatorname {Re}\Lambda ,$ $\operatorname {Im}\Lambda$ and their commutator are linearly independent, we show that $\mathcal {P}_\Lambda$ is not locally solvable, even in the presence of lower-order terms, provided that $n\ge 7$. 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 $\mathbb {H}_3$ a phenomenon first observed by Karadzhov and Müller in the case of $\mathbb {H}_2.$ 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.