Fractional Besov spaces and Hardy inequalities on bounded non-smooth domains

Abstract

Let \(\Omega \) be a bounded non-smooth domain in \(\mathbb {R}^n\) that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces \(B_{p,q}^s(\Omega )\), \(\mathring{B}_{p,q}^s(\Omega )\) and \(\widetilde{B}_{p,q}^s(\Omega )\) on \(\Omega \), which are defined, respectively, via the restriction, completion and supporting conditions with \(p,q\in [1,\infty )\) and \(s\in (0,1)\). The authors prove that \(B_{p,q}^s(\Omega )=\mathring{B}_{p,q}^s(\Omega )=\widetilde{B}_{p,q}^s(\Omega )\), if \(\Omega \) supports a fractional Besov–Hardy inequality, where the latter is proved under certain conditions on fractional Besov capacity or Aikawa’s dimension of the boundary of \(\Omega \).