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Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, be an open set with$s$-Ahlfors regular boundary $\partial \Omega$, for some $s \in(0,n]$, suchthat either $s=n$ and $\Omega$ is a corkscrew domain with the pointwise Johncondition, or $s<n$ and $\Omega= \mathbb{R}^{n+1} \setminus E$, for some$s$-Ahlfors regular set $E \subset \mathbb{R}^{n+1}$. In this paper we providea unifying method to construct Varopoulos' type extensions of $L^p$ and $BMO$boundary functions. In particular, we show that a) if $f \in L^p(\partial\Omega)$, $1<p\leq \infty$, there exists $F\in C^\infty(\Omega)$ such that thenon-tangential maximal functions of $F$ and $dist(\cdot, \Omega^c)|\nabla \barF|$ as well as the Carleson functional of $dist(\cdot,\Omega^c)^{s-n}\nabla F$are in $L^p(\partial \Omega)$ with norms controlled by the $L^p$-norm of $f$,and $F \to f$ in some non-tangential sense $\mathcal H^s|_{\partial\Omega}$-almost everywhere; b) if $\bar f\in BMO(\partial\Omega)$, there exists$\bar F\in C^\infty(\Omega)$ such that $dist(x, \Omega^c)|\nabla \bar F(x)|$ isuniformly bounded in $\Omega$ and the Carleson functional of$dist(x,\Omega^c)^{s-n}\nabla \bar F(x)$ as well the the sharp non-tangentialmaximal function of $\bar F$ are uniformly bounded on $\partial\Omega$ withnorms controlled by the ${BMO}$-norm of $\bar f$, and $\bar F \to \bar f$ in acertain non-tangential sense $\mathcal H^s|_{\partial \Omega}$-almosteverywhere. If, in addition, the boundary function is Lipschitz with compactsupport, then both $F$ and $\bar F$ can be constructed so that they are alsoLipschitz on $\overline\Omega$ and converge to the boundary data continuously.The latter results hold without the additional assumption of the pointwise Johncondition. Finally, we give some applications of the constructed extensions inthe connection between Poisson problems and BVPs.

Varopoulos extensions of boundary functions in $L^p$ and BMO in domains with Ahlfors-regular boundaries and applications