Dencker, Nils The solvability of non \(L^ 2\) solvable operators. (English) Zbl 0885.35151 Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 1996, Exp. No. 10, 11 p. (1996). 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) Cited in 1 ReviewCited in 2 Documents MSC: 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators Keywords:conjecture of Nirenberg; first-order pseudodifferential operators of principal type; condition \((\Psi)\); locally solvable Citations:Zbl 0818.35152; Zbl 0191.39103; Zbl 0208.35902 PDFBibTeX XMLCite \textit{N. Dencker}, Journ. Équ. Dériv. Partielles, St.-Jean-de-Monts 1996, Exp. No. 10, 11 p. (1996; Zbl 0885.35151) Full Text: Numdam EuDML