On Regularity and Extension of Green’s Operator on Bounded Smooth Domains

We prove regularity and extension results for Green’s operators that are associated to strictly elliptic second order divergence-type linear PDO’s with coefficients in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1,\alpha}(\overline{\Omega})$\end{document}. Here α ∈ (0, 1) and Ω ⊂ Rn, n ≥ 3, is a bounded C2,α domain. The regularity result gives boundary estimates for the derivatives up to order (2 + α) of the associated Green’s function. With the aid of this regularity result, we then extend the Green’s operator to a globally defined integral operator whose second order partial derivatives are Calderón–Zygmund singular integrals. We also show that, under reasonable a priori assumptions, the C2,α regularity of the domain is necessary for the aforementioned extension of the Green’s operator to a weakly singular integral operator, belonging to the class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\rm{SK}}^{-2}_{{\bf{R}}^n}(\alpha)$\end{document}.