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The resolution of the Nirenberg-Treves conjecture. (English) Zbl 1104.35080

This paper can be seen as the conclusion of a long series of articles, addressed to prove, or possibly disprove, the conjecture of Nirenberg-Trèves, which states the following. Let \(P= p(x,D)\) be a classical pseudodifferential operator of principal type, in the sense that \(p= 0\) implies \(d_\xi p\neq 0\), and assume condition \((\Psi)\): \(\text{Im}(ap)\) does not change sign from \(-\) to \(+\) along the oriented bicharacteristics of \(\text{Re}(ap)\), for any elliptic \(a\). Then \(P\) is locally solvable, namely the equation \(Pu= v\) has a local distribution solution \(u\) for every \(v\in C^\infty\).
When \(P\) is a linear partial differential operator, condition \((\Psi)\) simplifies and is necessary and sufficient for the local solvability, see for example L. Hörmander [The analysis of linear partial differential operators, I–IV, Springer Verlag, New York, 1983-85], where the reader finds also the proof of the necessity of \((\Psi)\) for pseudo-differential operators. A negative result was obtained by N. Lerner [Ann. Math. 128, 243–258 (1988; Zbl 0682.35112)], giving examples of pseudodifferential operators satisfying \((\Psi)\), which are not locally solvable with loss of 1 derivative in Sobolev estimates. In the present paper, the author proves that local solvability is always granted, under \((\Psi)\), if a loss of 2 derivatives is allowed.

MSC:

35S05 Pseudodifferential operators as generalizations of partial differential operators
47G30 Pseudodifferential operators
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)

Citations:

Zbl 0682.35112
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