Let Ω ⊂ R n+1 , n ≥ 1, be an open set with s-Ahlfors regular boundary ∂Ω, for some s ∈ (0, n], such that either s = n and Ω is a corkscrew domain with the pointwise John condition, or s < n and Ω = R n+1 \E, for some s-Ahlfors regular set E ⊂ R n+1 . In this paper we construct Varopoulos' type extensions of L p and BMO boundary functions. In particular, we show that a) if f ∈ L p (∂Ω), 1 < p ≤ ∞, there exists F ∈ C ∞ (Ω) such that the Carleson functional of dist(x, Ω c ) s−n ∇F (x) and the non-tangential maximal function of F are in L p (∂Ω) with norms controlled by the L p -norm of f , andwith norm controlled by the BMO-norm of f , and F → f in some non-tangential sense H s | ∂Ω -almost everywhere. If, in addition, the boundary functions are Lipschitz with compact support, then both F and F can be modified so that they are also Lipschitz on Ω and converge to the boundary data continuously. In the latter case, the pointwise John condition when s = n is only needed for the BMO extension.