Existence locale de solutions holomorphes pour les équations différentielles d'ordre infini (Q796755)

Let \(\Omega\) be an open neighbourhood of the origin and \(P=\sum a_{\alpha}(x)D^{\alpha}_ x\) be a partial differential operator of infinite order with holomorphic function \(a_{\alpha}(x)\) in \(\Omega\). One studies the surjectivity of P from \({\mathcal O}_ 0\) to \({\mathcal O}_ 0\) where \({\mathcal O}\) is the stalk of germs of holomorphic functions at 0. The author considers the case where a) every \(a_{\alpha}(x)\) is a polynomial of form \(\sum_{\alpha \leq \lambda}a^{\lambda}_{\alpha}x^{\lambda}\), b) every \(a_{\alpha}(x)\) is of the form \(x_{\alpha}\tilde a_{\alpha}(x)\) with a holomorphic function \(\tilde a{}_{\alpha}(x)\) on \(\Omega\), and establishes a sufficient condition for the surjectivity of operator in each case. In particular the author considers as an example an operator of the form \(P=\sum a_{\alpha}x^{\alpha}D^{\alpha}_ x\) with constant \(a_{\alpha}\) which is a generalization of the classical Euler operator in ordinary differential equation theory. In this case if one has the condition \(| \sum_{\alpha \leq \mu}a_{\alpha}/(\mu - \alpha)!| \geq r^{| \mu |}/\mu !\) with some r such that \(0<r<1\), then P is surjective from \({\mathcal O}_ x\) to \({\mathcal O}_ x\) for every x in \({\mathbb{C}}^ n\).