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A necessary and sufficient condition for Melin’s inequality for a class of systems. (English) Zbl 1007.35112
The authors give necessary and sufficient conditions for the Melin inequality to hold: Given a fixed formally selfadjoint $$N\times N$$ system $$A=A^{\text{ w}}(x,D_x)$$ of Weyl pseudodifferential operators of order $$m$$ on $${\mathbb R}^n$$, for any given $$\varepsilon>0$$ and any given compact $$K\subset{\mathbb R}^n$$ there exists a constant $$C=C_{K,\varepsilon}\in{\mathbb R}$$ such that $(Au,u)\geq-\varepsilon\|u\|^2_{(m-1)/2}-C\|u\|_{m/2-1}^2,\qquad \forall u\in C_0^\infty(K;{\mathbb C}^N).$ Following R. Brummelhuis [C. R. Acad. Sci., Paris, Sér. I 315, No. 2, 149-152 (1992; Zbl 0754.35197); Commun. Partial Differ. Equations 26, No. 9-10, 1559-1606 (2001; Zbl 1007.35111), preceding review], these conditions are given at each characteristic point, in terms of spectral properties of families of matrix oscillators. The proof of the sufficiency, however, requires an extra condition which is roughly stated by saying that the square of each derivative of the principal symbol has to be dominated, in the sense of Hermitian matrices, by the principal symbol itself. (It is not clear whether this extra condition is technical or not.) On the other hand, when the principal symbol has locally on the characteristic set constant rank $$r$$ and its determinant vanishes exactly to order $$2(N-r)$$ there, then only one suitable matrix oscillator has to be considered and the extra condition mentioned above is automatically fulfilled. In this way, since no condition is imposed on the symplectic nature of the characteristic set, the authors generalize the version of Melin’s inequality as proved by C. Parenti and A. Parmeggiani [J. Anal. Math. 86, 49-91 (2002; Zbl 1055.35153)] for systems with symplectic double characteristics. (However, the main result of that paper is an extension of L. Hörmander’s inequality [J. Anal. Math. 32, 118-196 (1977; Zbl 0367.35054)] to the case of systems.)
In the first section, after defining the crucial family of matrix oscillators, the authors state the main theorem, whose proof is given, by generalizing Melin’s arguments, in the second section. In the third and last section they show how to derive, in the case of systems of constant rank and precise order of vanishing on their characteristic set, a condition in terms of a single suitable matrix oscillator which is both necessary and sufficient.

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators
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##### References:
  R. G. M. Brummelhuis,Sur les inégalités de Gårding pour les systèmes d’opérateurs pseudodifférentiels, C. R. Acad. Sci. Paris, Sér. I315 (1992), 149–152. · Zbl 0754.35197  R. G. M. Brummelhuis,On Melin’s inequality for systems, Comm. Partial Differential Equations, to appear. (Earlier versions were available as CWI Report AM-R9111 (1991) and Univ. de Reims prépublication 97.07 (1997).)  L. Hörmander,The Cauchy problem for differential operators with double characteristics, J. Analyse Math.32 (1977), 118–196. · Zbl 0367.35054  L. Hörmander,The Analysis of Linear Partial Differential Operators III, Grundlehren Math. Wiss.274, Springer-Verlag, Berlin-Heidelberg, 1985.  A. Melin,Lower bounds for pseudo-differential operators, Ark. Mat.9 (1971), 117–140. · Zbl 0211.17102  J. Nourrigat,Subelliptic systems, Comm. Partial Differential Equations15 (1990), 341–405. · Zbl 0723.35089  J. Nourrigat,Systèmes sous-elliptiques II, Invent. Math.104 (1991), 377–400. · Zbl 0771.35086  C. Parenti and A. Parmeggiani,Lower bounds for systems with double characteristics, J. Analyse Math., to appear. · Zbl 1055.35153
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