Criteria of removable sets for harmonic functions in the sobolev spaces l1 p,w

Ahlfors and Beurling [16] proved that set is removable for class 𝐴𝐷2 of analytic functions with the finite Dirichlet integral if and only if does not change extremal distances. Their proof uses the conformal invariance of class 𝐴𝐷2, so it does not immediately generalize to 𝑝 ̸= 2 and to the relevant classes of harmonic functions in the space. In 1974 Hedberg [19] proposed new approaches to the problem of describing removable singularities in the function theory. In particular he gave the exact functional capacitive conditions for a set to be removable for class 𝐻𝐷𝑝(𝐺). Here 𝐻𝐷𝑝(𝐺) is the class of real-valued harmonic functions in a bounded open set ⊂ 𝑅𝑛, ≥ 2, and such that ∫︁ |∇𝑢|𝑝 1. In this paper we extend Hedberg's results on class 𝐻𝐷𝑝,𝑤(𝐺) of harmonic functions in and such that ∫︁ |∇𝑢|𝑝