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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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