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arXiv 2006.10682math.CA
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无Ahlfors正则性假设下有界调和函数的Carleson测度估计和$ε$ -逼近

Carleson measure estimates and $ε$-approximation of bounded harmonic functions, without Ahlfors regularity assumptions

John Garnett

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摘要：

令$Ω$为$\mathbb{R}^{d+1}$中的域$d \geq 1$。在文献[HMM 2]和[GMT]中证明了：若$Ω$满足螺旋条件，且$\partial Ω$是$d$-Ahlfors正则的，即对所有$x \in \partial Ω$和$0 < r < {\rm diam}(\partial Ω)$都是Hausdorff测度$\mathcal{H}^d(B(x,r) \cap \partial Ω) \sim r^d$，则$\partial Ω$是一致可求积的当且仅当（a）对于$Ω$上的每个有界调和函数，平方函数Carleson测度估计成立，或（b）$\varepsilon$ -所有$0 < \varepsilon <1$的近似性质。在这里，我们探讨（a）和（b）当$\partial Ω$不需要是Ahlfors规则的。我们首先证明了对于任何定义域$Ω$，存在一个定义域$\widetilde Ω\subset Ω$使得$\partial Ω\subset \partial \widetilde Ω$和$\partial \widetilde Ω$是一致可求积的，（a）和（b）成立.接下来，我们假设$Ω$满足螺旋形条件，并且$\partial Ω$满足容量密度条件。在这些假设下，我们反过来证明了这样的$\widetilde Ω$的存在性意味着（a）和（b）在$Ω$上成立，并给出了（a）或（b）成立的域的进一步特征。一个是调和测度满足Carleson packing条件，其直径类似于[GMT]中证明的电晕分解m等价于一致可求直性。第二个特征是让人想起的Carleson措施描述$H^{\infty}$插值序列的单位圆盘。

英文摘要：

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.

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