Traces and Extensions of Multipliers in Pairs of Sobolev Spaces

Abstract

It is well known that the space W ^l−1/p _p (R ⁿ⁻¹) with integer l is the space of traces on R ⁿ⁻¹ of functions in the Sobolev space W ^l _p (R ⁿ₊ ), where R ⁿ₊ = {(x, y): x ∈ R ⁿ⁻¹, y > 0 }. We show that a similar result holds for spaces of pointwise multipliers acting in a pair of Sobolev spaces. Namely, we prove that the traces on.R ⁿ of functions in the multiplier space M(W ^m _p (R ⁿ₊ ) → W ^l _p (R ⁿ₊ )) form the space M(W ^m−1/P _p (R ⁿ⁻¹) → W> ^l−1/P _p (R ⁿ⁻¹)), and that there exists a linear continuous extension operator which maps M(W ^m−1/P _p (R ⁿ⁻¹) → W ^l−1/P _p (R ⁿ⁻¹)) to M(W ^m _p (R ⁿ₊ ) → W ^l _p (R ⁿ₊ )). We apply this result to the Dirichlet problem for the Laplace equation in the half-space.