On certain imbedding and extension theorems for a weighted class of functions (Q1329204)

Let \(\Omega\) be a bounded domain in \(\mathbb{R}^n\) with boundary consisting of manifolds \(S\) and \(\Gamma\) whose dimensions are \(n- 1\) and \(m< n\) respectively. Let \(\rho_1(x)= \text{dist}(x, S)\) and \(\rho_2(x)= \text{dist}(x, \Gamma)\). Under suitable restrictions on the parameters, \(W^r_{p; \alpha_1, \alpha_2}(\Omega)\) is the class of functions \(f\) for which the norm \[ |f|_{L^p(\Omega)}+ \sum_{|k|= r} |\rho^{- \alpha_1}_1 \rho^{- \alpha_2}_2 D^k f|_{L^p(\Omega)} \] is finite. Embeddings of the form \[ W^r_{p; 0, \alpha_2}(\Omega)\to W^\lambda_{p; 0, \alpha_2+ r- \lambda}(\Omega)\to W^s_{p; 0, \alpha_2+ r-\lambda}(\Omega) \] are obtained, as are embeddings \(W^r_{p; 0, \alpha_2}(\Omega)\to B^{r+ \alpha_2}_p(\Omega)\) for non-weighted Besov spaces \(B^s_p(\Omega)\). Trace maps \(W^r_{p; 0, \alpha_2}(\Omega)\to B^{r+ \alpha_2- (n- m)/p}(\Gamma)\) and corresponding extensions are also discussed.