Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set

Let Ω be a John domain, let Γ ⊂ ∂Ω be an h-set, and let g and v be weights on Ω that are distance functions to the set Γ of special form. In the paper, sufficient conditions are obtained under which the Sobolev weighted class Wp,gr(Ω) is continuously embedded in the space Lq,v(Ω). Moreover, bounds for the approximation of functions in Wp,gr(Ω) by polynomials of degree not exceeding r − 1 in Lq,v(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \Omega $\end{document}) are found, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tilde \Omega $\end{document} is a subdomain generated by a subtree of the tree T defining the structure of Ω.