  We construct extensions of Varopolous type for functions $f \in
\text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More
precisely, let $\Omega \subset \mathbb{R}^{n+1}$ be an open set satisfying the
corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary
$\partial \Omega$, and let $\sigma := \mathcal{H}^n\lfloor_{\partial \Omega}$
denote the surface measure on $\partial \Omega$. We show that if $f \in
\text{BMO}(\partial \Omega,d\sigma)$ with compact support on $\partial \Omega$,
then there exists a smooth function $V$ in $\Omega$ such that $|\nabla V(Y)| \,
dY$ is a Carleson measure with Carleson norm controlled by the BMO norm of $f$,
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 $f \in C_c(\partial \Omega)$,
a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial \Omega)
\, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the
presence of an appropriate quantitative connectivity condition.

Analysis of PDEs;Classical Analysis and ODEs