Analytic Cauchy problem. I (Q1423888)

In this article the author studies singularities and analytic continuations of the Cauchy problem for linear partial differential operators with analytic coefficients in the complex domain. Let \(a(x,D)\) be a linear partial differential operator with analytic coefficients and \(g(x,D)\) the principal part of \(a(x,D)\) satisfying that \(g(x,1,0,\dots,0)=1\) in a neigborhood of the origin in \(\mathbb C^n\). Let \(v(x)\) be analytic in \(\{x\in \mathbb C^n; | x|\leq R,\;x_1\neq 0\}\) and \(u(x)\) analytic in \(\{x\in \mathbb C^n;| x |\leq R,\;x_1\neq 0\), \(\operatorname{Re} x_0>0\}\). He proves that if \(u\) satisfies the equation \(a(x,D)u=v(x)\) in a neigborhood of the origin, then \(u\) can be prolongated analytically in \(\{x\in \mathbb C^n;| x|\leq R_1,x_1\neq 0\}\).