论文标题
Carleson测量估算和$ε$ - 有限谐波功能的approximation，而无需AHLFORS规律性假设
Carleson measure estimates and $ε$-approximation of bounded harmonic functions, without Ahlfors regularity assumptions
论文作者
论文摘要
令$ω$为$ \ MATHBB {r}^{d+1} $，$ d \ geq 1 $中的一个域。在本文的参考文献中[hmm2]和[GMT]中，证明如果$ω$满足开瓶器状况，如果$ \ \ partial的ω$是$ d $ -d $ -Ahlfors常规，即Hausdorff Meature $ \ Mathcal {h} $ 0 <r <{\ rm diam}（\ partialω）$，然后$ \ partialω$在且仅当（a）平方函数carleson量度估计值均具有$ω$或（b）$ \ var parpsilon $ -approximation $ -approximation for $ var for $ var for $ var for y v var的情况下时，均可均匀地纠正。在这里，我们探索（a）和（b）当$ \ partialω$不需要ahlfors时。我们首先证明（a）和（b）对存在的任何域$ω$都保留，其中存在域$ \ wideTildeω\ subsetω$，使得$ \ partialω\ subset \ subset \ partial \ partial \ wideTildeω$和$ \ partial \ partial \ partial \ didetildeω$是均匀的。接下来，我们假设$ω$满足开瓶器条件，而$ \ partialω$满足容量密度的条件。在这些假设下，我们证明了这种$ \ widetildeω$的存在意味着（a）和（b）在$ω$上保留，并进一步表征（a）或（b）保留的域。一种是，谐波测量满足了与电晕分解相似的直径的Carleson填料条件，证明是[GMT]中均匀的可重新可透性。第二个表征是让人想起$ h^{\ infty} $插值序列的Carleson测量描述。
Let $Ω$ be a domain in $\mathbb{R}^{d+1}$, $d \geq 1$. In the paper's references [HMM2] and [GMT] it was proved that if $Ω$ satisfies a corkscrew condition and if $\partial Ω$ is $d$-Ahlfors regular, i.e. Hausdorff measure $\mathcal{H}^d(B(x,r) \cap \partial Ω) \sim r^d$ for all $x \in \partial Ω$ and $0 < r < {\rm diam}(\partial Ω)$, then $\partial Ω$ is uniformly rectifiable if and only if (a) a square function Carleson measure estimate holds for every bounded harmonic function on $Ω$ or (b) an $\varepsilon$-approximation property for all $0 < \varepsilon <1$ for every such function. Here we explore (a) and (b) when $\partial Ω$ is not required to be Ahlfors regular. We first prove that (a) and (b) hold for any domain $Ω$ for which there exists a domain $\widetilde Ω\subset Ω$ such that $\partial Ω\subset \partial \widetilde Ω$ and $\partial \widetilde Ω$ is uniformly rectifiable. We next assume $Ω$ satisfies a corkscrew condition and $\partial Ω$ satisfies a capacity density condition. Under these assumptions we prove conversely that the existence of such $\widetilde Ω$ implies (a) and (b) hold on $Ω$ and give further characterizations of domains for which (a) or (b) holds. One is that harmonic measure satisfies a Carleson packing condition for diameters similar to the corona decompositionm proved equivalent to uniform rectifiability in [GMT]. The second characterization is reminiscent of the Carleson measure description of $H^{\infty}$ interpolating sequences in the unit disc.
