Extension theorems of Whitney type by the use of integral operators

Abstract

Given a compact of ${\bf R}^n$, there is always a doubling measure having it as its support. We use this fact to construct an integral operator that extends differentiable functions defined on any compact set of ${\bf R}^n$ to the whole of ${\bf R}^n$. This allows us both to give a new proof of Whitney's extension theorem and to extend it to Besov spaces defined on arbitrary compact sets of ${\bf R}^n$. We also modify this operator to obtain, under certain assumptions, holomorphic extensions.