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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 $BMO$ and $L^p$boundary functions. In particular, we show that a) if $ f \in BMO(\partial\Omega)$, there exists $ F\in C^\infty(\Omega)$ such that $dist(x,\Omega^c)|\nabla F(x)|$ is uniformly bounded in $\Omega$ and the Carlesonfunctional of $dist(x,\Omega^c)^{s-n}|\nabla F(x)|$ as well the sharpnon-tangential maximal function of $ F$ are uniformly bounded on $\partial\Omega$ with norms controlled by the $BMO$-norm of $ f$, and $ F \to f$ in acertain non-tangential sense $\mathcal H^s|_{\partial \Omega}$-almosteverywhere; b) if $\bar f \in L^p(\partial \Omega)$, $1 <p \leq \infty$, thereexists $\bar F \in C^\infty(\Omega)$ such that the non-tangential maximalfunctions of $\bar F$ and $dist(\cdot, \Omega^c)|\nabla \bar F|$ as well as theCarleson functional of $dist(\cdot,\Omega^c)^{s-n}|\nabla \bar F|$ are in$L^p(\partial \Omega)$ with norms controlled by the $L^p$-norm of $\bar f$, and$\bar F \to \bar f$ in some non-tangential sense $\mathcal H^s|_{\partial\Omega}$-almost everywhere. If, in addition, the boundary function is Lipschitzwith compact support, then both $F$ and $\bar F$ can be constructed so thatthey are also Lipschitz on $\bar\Omega$ and converge to the boundary datacontinuously. The latter results hold without the additional assumption of thepointwise John condition. Finally, we give some applications of the constructedextensions in the connection between Poisson problems and BVPs.

Varopoulos extensions in domains with Ahlfors-regular boundaries and  applications to Boundary Value Problems for elliptic systems with $L^\infty$  coefficients