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Subelliptic systems. (English) Zbl 0723.35089

The author studies a system of 2p first order pseudo-differential operators \(X_ j(\lambda)\), \(j=1,...,2p\), depending on a parameter \(\lambda >1\). Setting \(L_ j(\lambda)=X_ j(\lambda)+iX_{p+j}(\lambda)\), \(j=1,...,p\), he obtains a necessary and sufficient condition for the estimate \[ \sum^{2p}_{j=1}\| X_ j(\lambda)f\|^ 2\leq C\sum^{p}_{j=1}\| L_ j(\lambda)f\|^ 2,\quad \forall f\in S'({\mathbb{R}}^ d), \] provided \(\lambda\) is a large parameter. The proof is based on a careful and deep analysis of some model operators for which the author uses some general previous results concerning nilpotent groups and described in the book “Hypoellipticité maximale pour des opérateurs polynômes de champs de vecteurs” (1985; Zbl 0568.35003) written jointly with B. Helffer. Some interesting applications for systems of subelliptic operators are given.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs

Citations:

Zbl 0568.35003
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References:

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