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


Authors: Detlef Müller and Marco M. Peloso
Journal: Trans. Amer. Math. Soc. 355 (2003), 2047-2064
MSC (2000): Primary 35A05, 35D05, 43A80
DOI: https://doi.org/10.1090/S0002-9947-02-03232-4
Published electronically: December 18, 2002
MathSciNet review: 1953537
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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

\begin{displaymath}\mathcal{P}_\Lambda= \sum_{i,j=1}^{n} \lambda_{ij}X_i Y_j={\,}^t X\Lambda Y, \end{displaymath}

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 $n\ge7$. 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 $\mathbb{H} _3$ 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.


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Additional Information

Detlef Müller
Affiliation: Mathematisches Seminar, C.A.-Universität Kiel, Ludewig-Meyn-Strasse 4, D-24098 Kiel, Germany
Email: mueller@math.uni-kiel.de

Marco M. Peloso
Affiliation: Dipartimento di Matematica, Corso Duca degli Abruzzi 24, Politecnico di Torino, 10129 Torino, Italy
Email: peloso@calvino.polito.it

DOI: https://doi.org/10.1090/S0002-9947-02-03232-4
Keywords: Local solvability, Heisenberg group
Received by editor(s): October 8, 2002
Published electronically: December 18, 2002
Additional Notes: We acknowledge the support for this work by the European Commission through the European TMR network “Harmonic Analysis" and the IHP Network HARP “Harmonic Analysis and Related Problems".
Article copyright: © Copyright 2002 American Mathematical Society

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