On regularity and extension of Green's operator on bounded smooth domains (Q438667)

The paper under review deals with regularity and extension results for Green's operators associated to divergence form uniformly elliptic second order linear differential operators with \(C^{1,\alpha}(\overline{\Omega})\)-regular coefficients, where \(\alpha \in (0, 1)\) and \(\Omega \subset {\mathbb R}^{n},\) \(n \geq 3,\) is a bounded domain with \(C ^{2,\alpha}\)-smooth boundary. The regularity result gives boundary estimates for the derivatives of the associated Green function up to order \((2+\alpha).\) By means of that regularity result, the author extends the Green operator to a globally defined integral operator whose second order partial derivatives are Calderón-Zygmund singular integrals. Under reasonable a~priori assumptions, it is also shown that the \(C ^{2,\alpha}\)-regularity of the domain \(\Omega\) is necessary for the extension of the Green operator to a weakly singular integral operator belonging to the class \({\text{SK}}^{-2}_{{{\mathbb R}}^n}(\alpha).\)