Traces of Sobolev Functions on Fractal Type Sets and Characterization of Extension Domains

Abstract

We describe traces of Sobolev functionsu∈W1, p(Rn), 1<p⩽∞, on certain subsets of Rnin terms of Sobolev spaces on metric spaces [7]. Our results apply to smooth submanifolds, fractal subsets, as well as to open subsets of Rn. In particular if 0⊂Rnis a John domain, then we characterize thoseW1, p(Ω) functions which can be extended toW1, p(Rn). IfΩis uniform, then this result implies Jones' extension theorem [14]. In the case of traces on fractal subsets our results are related to those of Jonsson and Wallin [16].