Embedding theorem for weighted Sobolev classes on a John domain with weights that are functions of the distance to some h-set
Anastasia Andreevna Vasil'eva · Russian Journal of Mathematical Physics · 2013
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 W (Ω) is continuously embedded in the space L q,v (Ω). Moreover, bounds for the approximation of functions in W (Ω) by polynomials of degree not exceeding r − 1 in L q,v ( $\tilde \Omega $ ) are found, where $\tilde \Omega $ is a subdomain generated by a subtree of the tree T defining the structure of Ω.