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

Abstract

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