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 a pointwise 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 lz; M (X → Y )l = lT ; X → Y l

in M (X → Y ). Let 1 ≤ p < ∞. Let m be an integer. W m

denotes the weighted Sobolev space with

m 1/p

1/p

the finite norm lulW m

p,ω0 ,ω1 l = lω0 |∇mu|lLp + lω1 ulLp,v . The aim of this work is to

obtain descriptions of multiplier spaces for the pair of weighted Sobolev spaces (W l

q,ω0 ,ω1 ).