Necessary and sufficient conditions for the local solvability of the Mizohata equations (Q583478)

The Mizohata equation \[ M_ nu(x_ 1,x_ 2)\equiv \partial u/\partial x_ 1+ix_ 1^{2n+1} \partial u/\partial x_ 2=f(x_ 1,x_ 2) \] does not have a distribution solution in any neighborhood of the origin for some \(C^{\infty}\) functions \(f(x_ 1,x_ 2)\). In this paper the author gives a necessary and sufficient condition on \(f(x_ 1,x_ 2)\) under which the equation \(M_ nu(x_ 1,x_ 2)=f(x_ 1,x_ 2)\) has a \(C^ 1\) solution in a neighborhood of the origin.