Fredholm property of partial differential operators of irregular singular type
For \(w= (w_1,w_2)\), \(w_j>0\), \(s>0\) let \(G^s_w\) be the Gevrey space of formal series \(u(x)=\sum_{k\geq 0}u_k{x^k\over k!}\) such that \(\sum_{k\geq 0}u_k{x^k\over(k!)^s}\) is holomorphic in the bidisk \(\{(|x_1|<w_1)\), \((|x_2|< w_2)\}\). Consider the differential operator \[ P(x,D)= \sum_{\beta\in\mathbb{Z}^2,\gamma\in\mathbb{N}^2} a_{\gamma\beta}D^\beta x^\gamma, \] a finite sum with respect to \(\beta\). Define the Toeplitz symbol associated with \(P(x,D)\) by \[ L_{s,w}(z,\xi)= \sum_{|\beta|+ (1-s)(|\alpha|- |\beta|)= m}a_{\alpha\beta}z^{\alpha-\beta} w^{\alpha-\beta}\xi^\alpha,\quad \xi\in\mathbb{R}^2, \] where the \(s\)-Gevrey order, \(m=\sup_{\gamma,\beta}[|\beta|+ (1-s)(|\gamma|- |\beta|),\;a_{\gamma\beta}\neq 0]\) of \(P(x,D)\) is supposed to be finite. The main result of the paper is the following: The operator \(P:G^s_w\to G^s_w\) is a Fredholm operator of index zero if the following condition is satisfied \[ L_{s,w}(z,\xi)\neq 0\quad\text{for} \quad \xi\geq 0,\;|\xi|=1\quad\text{and} \quad |z_1|= |z_2|=1. \] This result is strongly related to a similar result found in the 70s by Kashiwara-Kawai-Sjöstrand.
scientific article; zbMATH DE number 894904