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

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)

Zbl 0612.35001
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