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 $\mathrm{\Omega }\subset {\mathbb{R}}^{n+1}$ be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let $\sigma ≔{\mathcal{H}}^n\lfloor {}_{\partial \mathrm{\Omega }}$ denote the surface measure on ∂Ω. We show that if $f\in \text{BMO}(\partial \mathrm{\Omega },d\sigma )$ with compact support on ∂Ω, then there exists a smooth function V in Ω such that $|\mathrm{\nabla }V(Y)|dY$ 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 σ. 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 $f\in C_c(\partial \mathrm{\Omega })$, a harmonic extension u, with $|\mathrm{\nabla }u(Y){|}^2\text{dist}(Y,\partial \mathrm{\Omega })dY$ a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition.

我们为函数构造了 Varopolous 类型的扩展 $F\in \text{满地可}(乙)$对于任何均匀地可求组Ë余维之一。更准确地说，让$\mathrm{\Omega }\subset {\mathbb{电阻}}^{n+1}$是满足开瓶器条件的开集，具有n维一致可整边界 ∂Ω，令$\sigma ≔{\mathcal{H}}^n\lfloor {}_{\partial \mathrm{\Omega }}$表示 ∂Ω 上的表面测量。我们证明如果$F\in \text{满地可}(\partial \mathrm{\Omega },d\sigma )$在 ∂Ω 上有紧支持，则在 Ω 中存在光滑函数V使得$|\mathrm{\nabla }伏(是)|d是$是 Carleson 测度，Carleson 范数受f的 BMO 范数控制，并且V几乎在所有关于σ 的非切线意义上收敛到f。我们的结果应该与最近的几何特征进行比较${升}^{磷}$狄利克雷问题的可解性和 BMO 可解性，分别由 Azzam、第一作者、Martell、Mourgoglou 和 Tolsa 以及第一作者和 Le 撰写。结合起来，后一对结果表明，一个人可以构造，对于所有$F\in C_C(\partial \mathrm{\Omega })$,调和扩展u , 与$|\mathrm{\nabla }你(是){|}^2\text{区}(是,\partial \mathrm{\Omega })d是$一个Carleson测度与对Carleson规范通过的BMO规范受控˚F，仅在适当的定量的连接条件的存在。