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

Abstract

Let $\Omega \subset R^{n+1}$, $n\ge 1$, be an open set with s-Ahlfors regular boundary ∂Ω, for some $s\in (0,n]$, such that either $s=n$ and Ω is a corkscrew domain with the pointwise John condition, or $s<n$ and $\Omega =R^{n+1}\setminus E$, for some s-Ahlfors regular set $E\subset R^{n+1}$. In this paper we provide a unifying method to construct Varopoulos type extensions of BMO and $L^p$ boundary functions. In particular, we show that a) if $f\in \mathrm{BMO}(\partial \Omega )$, there exists $F\in C^{\infty }\left(\Omega \right)$ such that $\text{dist}(x,{\Omega }^c)\left|\nabla F\right(x\left)\right|$ is uniformly bounded in Ω and the Carleson functional of $\text{dist}{(x,{\Omega }^c)}^{s-n}\left|\nabla F\right(x\left)\right|$ as well the sharp non-tangential maximal function of F are uniformly bounded on ∂Ω with norms controlled by the BMO-norm of f, and $F\to f$ in a certain non-tangential sense $H^s{|}_{\partial \Omega }$-almost everywhere; b) if $\overline{f}\in L^p(\partial \Omega )$, $1<p\le \infty$, there exists $\overline{F}\in C^{\infty }\left(\Omega \right)$ such that the non-tangential maximal functions of $\overline{F}$ and $\text{dist}(\cdot ,{\Omega }^c)\left|\nabla \overline{F}\right|$ as well as the Carleson functional of $\text{dist}{(\cdot ,{\Omega }^c)}^{s-n}\left|\nabla \overline{F}\right|$ are in $L^p(\partial \Omega )$ with norms controlled by the $L^p$-norm of $\overline{f}$, and $\overline{F}\to \overline{f}$ in some non-tangential sense $H^s{|}_{\partial \Omega }$-almost everywhere. If, in addition, the boundary function is Lipschitz with compact support, then both F and $\overline{F}$ can be constructed so that they are also Lipschitz on $\bar{\Omega }$ and converge to the boundary data continuously. The latter results hold without the additional assumption of the pointwise John condition. Finally, for elliptic systems of equations in divergence form with merely bounded complex-valued coefficients, we show some connections between the solvability of Poisson problems with interior data in the appropriate Carleson or tent spaces and the solvability of Dirichlet problem with $L^p$ and BMO boundary data.