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The solvability of non \(L^ 2\) solvable operators. (English) Zbl 0885.35151

The author considers the counterexamples of N. Lerner [Ann. Math. 139, No. 2, 363-393 (1994; Zbl 0818.35152)], about first-order pseudodifferential operators of principal type satisfying condition \((\Psi)\), which are not locally solvable in \(L^2\). The author shows that such operators are actually locally solvable for \(C^\infty\) data, according to the conjecture of L. Nirenberg and F. Trèves [Commun. Pure Appl. Math. 23, 1-38 (1970; Zbl 0191.39103) and ibid. 459-509 (1970; Zbl 0208.35902)], which claims that \((\Psi)\) is equivalent to local solvability for principal type.
As is well known, this conjecture is true for partial differential equations; the present discussions refer to the pseudodifferential case.
Reviewer: L.Rodino (Torino)

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
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