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

Let \(\Omega \) $\mathrm{\Omega }$ be a bounded non-smooth domain in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ that satisfies the measure density condition. In this paper, the authors study the interrelations of three basic types of Besov spaces \(B_{p,q}^s(\Omega )\) $B_{p,q}^s(\mathrm{\Omega })$ , \(\mathring{B}_{p,q}^s(\Omega )\) ${\mathring{B}}_{p,q}^s(\mathrm{\Omega })$ and \(\widetilde{B}_{p,q}^s(\Omega )\) ${\tilde{B}}_{p,q}^s(\mathrm{\Omega })$ on \(\Omega \) $\mathrm{\Omega }$ , which are defined, respectively, via the restriction, completion and supporting conditions with \(p,q\in [1,\infty )\) $p,q\in [1,\infty )$ and \(s\in (0,1)\) $s\in (0,1)$ . The authors prove that \(B_{p,q}^s(\Omega )=\mathring{B}_{p,q}^s(\Omega )=\widetilde{B}_{p,q}^s(\Omega )\) $B_{p,q}^s(\mathrm{\Omega })={\mathring{B}}_{p,q}^s(\mathrm{\Omega })={\tilde{B}}_{p,q}^s(\mathrm{\Omega })$ , if \(\Omega \) $\mathrm{\Omega }$ 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 \(\Omega \) $\mathrm{\Omega }$ .