A necessary and sufficient condition for Melin's inequality for a class of systems
type: Item
instance of: scholarly article
title: A necessary and sufficient condition for Melin's inequality for a class of systems
zbMATH Open document ID: 1007.35112
DOI: 10.1007/BF02788080
published in: Journal d'Analyse Mathématique
publication date: 2002-07-10T00:00:00Z
review text: 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 \textit{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 \textit{C. Parenti} and \textit{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 \textit{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 …
reviewed by: Alberto Parmeggiani
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 1765845
zbMATH Keywords: lower bounds
zbMATH Keywords: necessary and sufficient conditions for the Melin inequality
zbMATH Keywords: Weyl pseudodifferential opeators
zbMATH Keywords: matrix oscillators
author: R. G. M. Brummelhuis
author: J. Nourrigat
MaRDI profile type: Publication
cites work: Q4014794
cites work: The Cauchy problem for differential equations with double characteristics
cites work: Lower bounds for pseudo-differential operators
cites work: Subelliptic systems
cites work: Systèmes sous-elliptiques. II. (Subelliptic systems. II)
cites work: Lower bounds for systems with double characteristics.
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description: scientific article; zbMATH DE number 1765845