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

Abstract

Let 1 ≤ p < ∞ and ω be Beurling's weight function on ℝd. In this article, we demonstrate some harmonic properties of Sobolev space Wp<inf>k</inf> (ℝd) which consists of all functions on ℝd whose weak derivatives up to and including order k belong to Lp(ℝd). Also, we deal with some harmonic properties of intersection space Ap<inf>k,ω</inf>(ℝd) = L1<inf>ω</inf>(ℝd) ∩ Wp<inf>k</inf> (ℝd) by aid of Beurling algebra L1<inf>ω</inf>(ℝd) and Sobolev space Wp<inf>k</inf> (ℝd). Finally, we research the inclusions and continuous embeddings between the spaces Ap<inf>k,ω</inf>(Ω) where Ω ⊂ ℝd be open set. © 2015 Ramanujan Mathematical Society.