Let u be an Ap weight on ℝn+1 and v a doubling weight on ℝn. Define the trace, or restriction, operator Trf(x (′) = f(x =′, 0), where x′ ∈ ℝn and f is a function on ℝn+1. If α >1p+n1 p−1, where β is the+β−n + p doubling exponent of v, then the trace operator is bounded from the weighted Bessel potential space Lpα(u) (which coincides with the weighted Sobolev space Lp p k(u) if α = k ∈ N) into the weighted Besov space Bα−1/p,p (v) if and only if there exists C > 0 such that ∫ 1 v dx′ ≤ C1 ∫ udx |I| I |Q(I)| Q(I) for all dyadic cubes I ⊆ ℝn with side length less than or equal to 1, where Q(I) = I×[0, ℓ(I)]. If u and v satisfy the converse inequality, then there exists a continuous linear map Ext: Bpα−1/p, p (v) → Lpα(u). If both inequalities hold, Tr ◦ Ext is the identity on Bα−1/p, p p (u). More generally, the results hold with (u) for any Lpα(u) replaced by the inhomogeneous Triebel-Lizorkin space Fpα, q 0 < q ≤ ∞.
