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Non-solvability for a class of left-invariant second-order differential operators on the Heisenberg group. (English) Zbl 1072.35010

Summary: 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 \(n\geq7\). 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 \(3\leq n\leq 6\) the analysis of these operators seems to become quite complex, from a technical point of view, and it remains open at this time.

MSC:

35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35D05 Existence of generalized solutions of PDE (MSC2000)
43A80 Analysis on other specific Lie groups
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