论文标题
统一的重新讨论性意味着Varopoulos扩展
Uniform rectifiability implies Varopoulos extensions
论文作者
论文摘要
对于任何均匀整流的condimension One的$ f \ in \ text {bmo}（e）$，我们为功能$ f \构建了Varopolos类型的扩展。更准确地说，令$ω\ subset \ mathbb {r}^{n+1} $为满足开瓶器条件的开放式集合，具有$ n $ dimensional均匀地统一的边界$ \ partialω$，并让$ = $σ：= = \ nation calcal { $ \部分ω$。我们表明，如果$ f \ in \ text {bmo}（\ partialω，dσ）$在$ \ partialω$上进行紧凑的支持，则存在$ω$中的平滑函数$ v $，以便$ | \ nabla v（y）| \，dy $是Carleson措施，由Carleson Norm由$ F $的BMO规范控制，因此，$ v $以某些非不同意义收集到$ f $几乎到处都在$σ$方面。我们的结果应与$ l^p $ - 可辨方差的最新几何特征和Dirichlet问题的BMO可溶性相比，由Azzam，第一作者，Martell，Martell，Mourgoglou和Tolsa以及第一作者和LE。 In combination, this latter pair of results shows that one can construct, for all $f \in C_c(\partial Ω)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial Ω) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.
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 $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ 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 $σ$. 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 $f \in C_c(\partial Ω)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial Ω) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.
