Uniform rectifiability implies Varopoulos extensions

We construct extensions of Varopolous type for functions f ∈ BMO ( E ) , any uniformly rectifiable set codimension one. More precisely, let Ω ⊂ R n + 1 be an open satisfying the corkscrew condition, with -dimensional boundary ∂Ω, and σ ≔ H ⌊ ∂ denote surface measure on ∂Ω. show that if d compact support then there exists a smooth function V in such | ∇ Y is Carleson norm controlled by converges some non-tangential sense to almost everywhere respect . Our results should compared recent geometric characterizations L p -solvability BMO-solvability Dirichlet problem, Azzam, first author, Martell, Mourgoglou Tolsa author Le, respectively. In combination, this latter pair shows one can construct, all C c harmonic extension u 2 dist only presence appropriate quantitative connectivity condition.