On multipliers in weighted Sobolev spaces. Part I

Abstract

Let X, Y be Banach spaces whose elements are functions y : Ω → R. We say that a function z : Ω → R is apointwise multiplier on the pair (X, Y ), if T x = zx ∈ Y and the operator T : X → Y is bounded. M(X → Y )denotes the multiplier space on the pair (X, Y ). We introduce the norm kz; M (X → Y )k = kT ; X → Y kin M (X → Y ). Let 1 ≤ p < ∞. Let m be an integer. Wmp,ω0,ω1denotes the weighted Sobolev space withthe finite norm kukWmp,ω0,ω1= ku; Wmp,ω0,ω1k = kω1/p0|∇mu|kLp+ kω1/p1ukLp,v. The aim of this work is toobtain descriptions of multiplier spaces for the pair of weighted Sobolev spaces (Wlp,ρ,v, Wmq,ω0,ω1).