The Gakhov decomposition for pseudodifferential operators with degenerate symbol (Q1972648)

The author studies solvability of the equation \(Af=g\), where \(f\) and \(g\) are functions on a~two-dimensional surface and \(A\) is a~pseudodifferential operator with degenerate symbol. The author describes conditions for a two-dimensional torus implying the representation \(A=C\cdot Q_\varkappa\cdot B +T_r \), where the operator \(T_r \) is completely continuous, the operators~\(C\) and~\(B\) are invertible, and a~solution to the equation \(Q_\varkappa f=g\) can be constructed straightforward. Moreover, conditions are given for solvability of the ``regularized'' equation \((A-T_r)f=g\).