A note on Hajłasz-Sobolev spaces on fractals
Let \(M \subset\mathbb{R}^n\) and let \(\mu\) be a Borel measure on \(M\). Then \(L_p (M,\mu)\) with \(1\leq p \leq \infty\) has the usual meaning. For \(\sigma >0\) the Hajłasz-Sobolev space \(H^\sigma_p (M,\mu)\) consists of all \(u \in L_p (M, \mu)\) such that \[ |u(x) - u(y)|\leq |x-y|^\sigma (v(x) + v(y)) \] for some \(v \in L_p (M, \mu)\). (\(\sigma >1\) makes sense if, for example, \(M\) is a disconnected fractal.) It is the main aim of this paper to compare these spaces with the well-known spaces \(B^\sigma_{p, \infty} (M, \mu)\) introduced by Jonsson and Wallin. In addition, some embedding theorems are proved.
scientific article; zbMATH DE number 1915607