Uniform rectifiability implies Varopoulos extensions

We construct extensions of Varopolous type for functions $f \in $\text{BMO}$(E)$,foranyuniformlyrectifiableset$E$ of codimension one. More precisely, let $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\mathrm{\partial }\mathrm{\Omega }$, and let $\sigma :={\mathcal{H}}^n{\lfloor }_{\mathrm{\partial }\mathrm{\Omega }}$ denote the surface measure on $\mathrm{\partial }\mathrm{\Omega }$. We show that if $f \in $\text{BMO}$(\partial \Omega,d\sigma)$withcompactsupporton$\partial \Omega$, then there exists a smooth function $V$ in $\mathrm{\Omega }$ such that $|\nabla V(Y)| , dY$isaCarlesonmeasurewithCarlesonnormcontrolledbytheBMOnormof$f$, and such that $V$ converges in some non-tangential sense to $f$ almost everywhere with respect to $\sigma$. Our results should be compared to recent geometric characterizations of $L^p$-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all $f\in C_c(\mathrm{\partial }\mathrm{\Omega })$, a harmonic extension $u$, with $|\nabla u(Y)|^2 $\text{dist}$(Y,\partial \Omega) , dY $aCarlesonmeasurecontrolledbytheBMOnormof$f$, only in the presence of an appropriate quantitative connectivity condition.