Traces and Extensions of Multipliers in Pairs of Sobolev Spaces

Abstract

It is well known that the space W p l−1/p (R n−1) with integer l is the space of traces on R n−1 of functions in the Sobolev space W p l (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 p m (R + n ) → W p l (R + n )) form the space M(W p m−1/P (R n−1) → W> p l−1/P (R n−1)), and that there exists a linear continuous extension operator which maps M(W p m−1/P (R n−1) → W p l−1/P (R n−1)) to M(W p m (R + n ) → W p l (R + n )). We apply this result to the Dirichlet problem for the Laplace equation in the half-space.