Sufficiency of condition (\(\psi\) ) for local solvability in two dimensions (Q1824118)

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 \textit{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.