On the Whitney problem for weighted Sobolev spaces

Аннотация

Given a closed weakly regular d-thick subset S of ℝⁿ, we prove the existence of a bounded linear extension operator Ext: Tr_|SW_p¹ (ℝⁿ, γ) → W_p¹(ℝⁿ, γ) for p ∈ (1, ∞), 0 ≤ d ≤ n, r ∈ (max{1, n − d}, p), l ∈ ℕ, and \(\gamma \in {A_{\frac{p}{r}}}\)(ℝⁿ). In particular, we prove that a linear bounded trace space exists in the case where S is the closure of an arbitrary domain in ℝⁿ, γ ≡ 1, and p > n − 1. The obtained results supplement those of previous studies, in which a similar problem was considered either in the case of p ∈ (n, ∞) without constraints on the set S or in the case of p ∈ (1, ∞) under stronger constraints on the set S.