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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)

##### MSC:
 35S05 Pseudodifferential operators as generalizations of partial differential operators 47G30 Pseudodifferential operators
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