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Nonsolvability in \(L^ 2\) for a first order operator satisfying condition \((\psi)\). (English) Zbl 0818.35152
The author proves that condition \((\psi)\) is not sufficient for the \(L^ 2\)-local solvability of pseudo-differential operators. We recall that, for a classical pseudo-differential operator \(P\) with homogeneous principal symbol \(p(x, \xi)\) of principal type, condition \((\psi)\) means that \(\text{Im} p\) does not change sign from \(-\) to \(+\) along the oriented bicharacteristic curves of \(\text{Re} p\). L. Nirenberg and F. Trèves [Commun Pure Appl. Math. 23, 1-38 (1970; Zbl 0191.391) and 459-509 (1970; Zbl 0208.359)] conjectured that \((\psi)\) is equivalent to local solvability of \(P\) and proved it in a number of particular cases. The general proof of the necessity of \((\psi)\) can be found in L. Hörmander [The analysis of linear partial differential operators IV. (1985; Zbl 0612.35001)]. The conjecture of sufficiency is here disproved, in some sense, by means of an example in \(\mathbb{R}^ 3\) of the form \[ P = D_ t + iQ(t,x, D_ x) \] where \(Q\) is a first order pseudo-differential operator with real principal symbol \(q(t,x, \xi)\), \(x = (x_ 1, x_ 2)\), \(\xi = (\xi_ 1, \xi_ 2)\). Precisely \(q\) is constructed satisfying \((\psi)\) such that \(Pu = f\) has no \(L^ 2\) solution for a general right-hand side \(f\) in \(L^ 2\). The proof is technical, but the author explains the main ideas in an interesting introduction.
Let us observe that one deals in this paper with somewhat exotic pseudo- differential operators, not connected with true differential problems; in fact, for partial differential operators, \((\psi)\) can be replaced by the so-called condition \((P)\), which characterizes local solvability, cf. R. Beals and C. Fefferman [Ann. Math., II. Ser. 97, 482-498 (1973; Zbl 0256.35002)].
Reviewer: L.Rodino (Torino)

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