On local solvability of pseudodifferential operators with double characteristics (Q1117410)

Consider the equation \(P(x,D)u=f(x)\) where P(x,D) is a pseudodifferential operator with double characteristics and \(f(x)\in C_ 0^{\infty}(\Omega)\), \(\Omega \subset R^ n.\) We say, the operator P(x,D) is locally solvable at a point \(x^ 0\in \Omega\), if there exists a neighbourhood of \(x^ 0\), \(\omega\) \(\subset \Omega\), in which for any function \(f\in C_ 0^{\infty}(\omega)\) there is a distribution \(u\in E'(\Omega)\) such that \(P(x,D)u=f(x)\) in \(\omega\). Necessary conditions are introduced for the local solvability of the pseudodifferential operator.