Lichawski-Matkowski-Mi theorem on locally defined operators for functions of several variables

Let $D$ be a regular closed set in the open subspace $G subset mathbb{R}^{n} R$ and $C^{m} (D)$ be the space of functions $f|_{D}$ such that $f in C^{m} (G)$. The representation formulas for locally defined operators mapping $C^{m} (D)$ into $C^{0} (D)$ and into $C^1 (D)$ are given.