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Generalizations of Melin’s inequality to systems. (English) Zbl 1054.35153
Proceedings of the conference on partial differential equations, Plestin-les-Grèves, France, June 5–8, 2001. Exp. Nos. I-XIV. Nantes: Université de Nantes (ISBN 2-86939-169-2/pbk). Exp. No. 1, 8 p. (2001).
This is mainly an expository article that provides insight in the difficult problem of finding lower bounds for systems of pseudo-differential operators. Specifically, the article concentrates on the problem of finding roughly necessary and sufficient conditions for Melin’s inequality to hold in the case of systems with double characteristics. The author specially discusses results of R. Brummelhuis and J. Nourrigat [J. Anal. Math. 85, 195–211 (2001; Zbl 1007.35112)] that improve results of C. Parenti and A. Parmeggiani [J. Anal. Math. 86, 49–91 (2002; Zbl 1055.35153)]. Specifically, the assumption that the characteristic set must be a symplectic manifold is eliminated. The author comments at length the value of various hypotheses.
The paper ends with a short section on proofs, where a few items from the proofs of the main results in the paper by R. Brummelhuis and J. Nourrigat are highlighted.
For the entire collection see [Zbl 0990.00046].
MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B45 A priori estimates in context of PDEs
47G30 Pseudodifferential operators
53D05 Symplectic manifolds (general theory)
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