Meromorphic representations of the solutions of the singular Cauchy problem. I (Q1093090)

Consider a partial differential operator L of the form \[ L=D^ 2_ t- xD^ 2_ x-cD_ x+\tilde c(t,x)D_ x+a(t,x)D_ t+b(t,x), \] where a(t,x), b(t,x) and \(\tilde c(t,x)\) are holomorphic functions defined in a neighborhood U of the origin of \(C^ 2\). The author considers the Cauchy problem \(Lu=0\), \(u(0,x)=w_ 1(x)\), \(D_ tu(0,x)=w_ 2(x)\), where at least one of the \(w_ 1\) respectively \(w_ 2\) has a pole at the origin and both are holomorphic for \(x\neq 0\) in a small neighborhood of the origin. It is shown that for a suitable surface K, constructed from (complex) characteristic surfaces of L and for a suitable complex neighborhood U' of 0 the Cauchy problem from the above admits a holomorphic solution on the universal covering space over \(U\setminus K\).