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Perturbation and energy estimates. (English) Zbl 0927.35139
The paper is devoted to the problem of the sufficiency of the \((\psi)\) condition of Nirenberg-Trèves for the local solvability of the pseudodifferential operators of principal type; see on the subject the recent paper of N. Lerner [Ann. Math., II. Ser. 139, No. 2, 363-393 (1994; Zbl 0818.35152)]. Here, the author proves the following. Let \(P\) be a pseudodifferential operator with symbol in \(S^1_{1,0}\); assume it admits a principal part satisfying condition \((\psi)\). Then there exists an \(L^2\)-bounded operator \(R\) such that the equation \((P+R)u= f\) has a local solution in \(H^{-1}\) for \(f\) in \(L^2\). In general \(R\) has to be taken with non-classical symbol in \(S^0_{1/2,1/2}\).
The paper is written in a very clear way and, despite the involved technicality, it is readable by non-experts.
We observe the following. It is known that there exists \(P\) satisfying \((\psi)\), for which \(Pu= f\in L^2\), or \((P+R)u= f\in L^2\), do not admit \(L^2\) solutions; that is, a loss of regularity \(u\in H^{-1}\) is expected in general. It is likely, in view of previous counterexamples, that there exists \(P\) satisfying \((\psi)\), for which \((P+R)u= f\in L^2\) do not admit solutions \(u\in H^{-1-N}\), \(N>0\), if the \(L^2\) bounded operator \(R\) is arbitrary; that is, one has to be very careful in choosing \(R\), to obtain \(u\in H^{-1}\). A natural guess is that for every \(P\) satisfying \((\psi)\) and for every \(N\) there exists \(R_N\) regularizing of order \(N\) such that \((P+R_N)u= f\in L^2\) admits solutions \(u\in H^{-1-N}\).
Reviewer: L.Rodino (Torino)

MSC:
35S05 Pseudodifferential operators as generalizations of partial differential operators
35A27 Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs
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