Detlef Müller, Marco M. Peloso, Fulvio Ricci:

On local solvability for complex coefficient differential operators on the Heisenberg group

We give several examples of operators that are not locally solvable for all choices of certain parameters, even if one allows the addition of lower order terms, in some cases also non-invariant ones. This is in striking contrast with the phenomenona known so far in the theory of local solvability of invariant second-order differential operators on nilpotent Lie groups.

In order to disprove local solvability we use two different technical tools. The first one is a criterion by Hörmander [1]. The second one is an extension of a criterion for local solvability in [2]. This extension, which is of interest in its own right, allows us to deal with non-homogenous invariant differential operators.

Our analysis of the differential operators is based on the classification of normal forms for involutive complex Hamiltonians under the action of the real symplectic group [3].

[1] L. Hörmander:

Linear Partial Differential Operators. Springer-Verlag, Berlin, 1964.

[2] L. Corwin, L. P. Rothschild:

Necessary conditions for local solvability of homogeneous left invariant operators on nilpotent Lie groups. Acta Math. 147, pp. 265-288, 1981.

[3] D. Müller, C. Thiele:

Normal forms for involutive complex Hamiltonian matrices under the real symplectic group. Preprint, 1997.

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

Bibliographical note: J. Reine Angew. Math. 513 (1999), 181--234

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[Thu Feb 19 18:56:34 2009]

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