Sobolev spaces on non-smooth domains and Dirichlet forms related to subordinate reflecting diffusions (Q2721355)

The paper under review may be considered as a sequel to the paper [\textit{N. Jacob} and \textit{R. L. Schilling}, Rev. Mat. Iberoam. 15, 59-91 (1999; Zbl 0922.31011)]. The subject of the paper is again the interplay between function spaces and interpolation, subordination in the sense of Bochner, and Dirichlet forms. The difference is that in the former paper the authors considered smooth domains, while in the present paper the emphasis is on non-smooth domains. NEWLINENEWLINENEWLINEThe first part of the paper is devoted to a detailed introduction to Sobolev spaces over non-smooth domains and to proving some new results such as the trace theorems. More precisely, let \(\Omega\) be a bounded \((\varepsilon, \delta)\) domain in \({\mathbb{R}}^n\) with the boundary \(\partial \Omega\) that is a \(d\)-set with \(n-1\leq d <n\). For \(s\in {\mathbb{R}}\) and \(1<p<\infty\) such that \((n-d)/p<s<1+(n-d)/p\), the trace operator NEWLINE\[NEWLINE\text{tr}_{\partial \Omega}:H^s_p(\Omega) \to B^{s-(n-d)/p}_{pp}(\partial \Omega)NEWLINE\]NEWLINE is well-defined, and it is a bounded linear surjection with the bounded right inverse. Here \(H^s_p(\Omega)\) is the Sobolev space on \(\Omega\), and \(B^{s-(n-d)/p}_{pp}(\partial\Omega)\) the Besov space on \(\partial \Omega\). The authors prove that for the case \((n-d)/p<s\leq 1\), the kernel of the trace operator \(\text{tr}_{\partial \Omega}\) is the space \(H_p^{0,s}(\Omega)\) (i.e. the closure of \(C_0^{\infty}(\Omega)\) in the norm of \(H_p^s(\Omega)\)), while in the case \(0<s<(n-d)/p\) it holds that \(H_p^s(\Omega)=H_p^{0,s}(\Omega)\).NEWLINENEWLINENEWLINEIn the second part of the paper the authors obtain the Weyl decomposition of the space \(H^{\alpha}(\Omega):=H^{\alpha}_2(\Omega)\) as the orthogonal sum (with respect to an appropriate inner product) of \(H^{0,\alpha}(\Omega)\) and the space of harmonic functions. Due to a result from the first part, this decomposition is nontrivial only in the case when \(\alpha > (n-d)/2\). In fact, \(H^{\alpha}(\Omega)\) is identified as the domain of a subordinate Dirichlet form. By use of the Weyl decomposition, the restriction of that form to the boundary is shown to be equivalent to an analog of the classical Douglas integral.