On the weighted equivalence of open sets in Rn

Ahlfors and Beurling gave a characterization in terms of extremal distances of the removable singularities for the class of analytic functions with finite Dirichlet integral. Following Ahlfors and Beurling refer a relatively closed set contained in open set ⊂ as an 𝑁𝐶𝑝,𝑤-set if do not affect the(𝑝,𝑤) modulus 𝑚𝑝,𝑤(𝐹0, 𝐹1, Π) for every coordinate rectangle Π ⊂ 𝐺. Dymchenko and Shlyk established that 𝑁𝐶𝑝,𝑤-sets are removable for the weighted Sobolev space 𝐿1 𝑝,𝑤(𝐺). Observe that the idea to study removable sets of this type in 𝑅𝑛, ≥ 2, in terms of rectangle is not new and for ≡ 1 was considered by Hedberg, Yamamoto. In particular Hedberg gave the definition of set ⊂ Π for a certain condenser capacity and showed that such set is removable for the class of real valued harmonic function with vanishing periods, ∫︀ |∇𝑢|𝑝