Sufficiency of condition (\(\psi\) ) for local solvability in two dimensions. (English) Zbl 0682.35112

Lerner, Nicolas

Ann. Math. (2) 128, No. 2, 243-258 (1988).

The author proves the sufficiency of condition \(\psi\) for local solvability in two dimensions; precisely: local solutions of the equation \(Pu=f\) exist, if P is any classical pseudodifferential operator in two dimensions, of principal type, and the imaginary part b of the principal symbol of P does not change sign from - to \(+\) along any oriented bicharacteristic of the real part a of the principal symbol.
Condition \(\psi\) is necessary for local solvability in general, as proved in the book of L. Hörmander [The analysis of linear partial differential operators. IV: Fourier integral operators (1985; Zbl 0612.35001)]. The problem of the sufficiency of the condition remains open in higher dimensions.

Reviewer: L.Rodino

MSC:

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

Keywords:

sufficiency of condition \(\psi \); two dimensions; bicharacteristic; principal symbol

Citations:

Zbl 0612.35001