A Note on Weighted Sobolev Algebras L1ω(ℝd) ∩ Wpk (ℝd)

Let 1≤p<∞ and ω be Beurling’s weight function on ℝ^d. In this article, we demonstrate some harmonic properties of Sobolev space W^p_k (ℝ^d) which consists of all functions on ℝ^d whose weak derivatives up to and including order k belong to L^p(ℝ^d). Also, we deal with some harmonic properties of intersection space A^p_k,ω(ℝ^d) = L¹_ω(ℝ^d)∩W^p_k (ℝ^d) by aid of Beurling algebra L¹_ω(ℝ^d) and Sobolev space W^p_k (ℝ^d). Finally, we research the inclusions and continuous embeddings between the spaces A^p_k,ω(Ω) where Ω⊂ℝ^d be open set.