满足容量密度条件的单边NTA域上椭圆算子的平方函数和非切极大函数估计
Square function and non-tangential maximal function estimates for elliptic operators in 1-sided NTA domains satisfying the capacity density condition
摘要：
设$Ω\subset\mathbb{R}^{n+1}$，$n\ge 2$是一个单侧非切向可达域（也称为一致域），即$Ω$满足内部Corkscrew和Harnack链条件，这两个条件分别是开放性和路径连通性的标度不变/定量版本.我们还假设$Ω$满足所谓的容量密度条件，这是所有边界点都是维纳正则的事实的定量版本。考虑$L_0 u=-\mathrm{div}(A_0\nabla u)$，$Lu=-\mathrm{div}(A\nabla u)$，$Ω$中的两个实（非必要对称）一致椭圆算子，并将相应的相关椭圆测度写为$ω_{L_0}$，$ω_L$。该程序的目标是找到保证$ω_L$满足关于$ω_{L_0}$的$A_\infty$ -条件或$RH_q$ -条件的充分条件。在本文中，我们感兴趣的是获得平方函数和非切估计的了解以前。我们建立了有界弱零解满足Carleson测度估计，关于相关的椭圆测度。我们还证明了，对于每个弱零解，相关的平方函数可以控制的非切极大函数在任何Lebesgue空间关于相关的椭圆测度。这些结果扩展了Dahlberg-Jerison-Kenig的工作，并且是arXiv：1901.08261中微扰结果的证明的基础.
英文摘要：
Let $Ω\subset\mathbb{R}^{n+1}$, $n\ge 2$, be a 1-sided non-tangentially accessible domain (aka uniform domain), that is, $Ω$ satisfies the interior Corkscrew and Harnack chain conditions, which are respectively scale-invariant/quantitative versions of openness and path-connectedness. Let us assume also that $Ω$ satisfies the so-called capacity density condition, a quantitative version of the fact that all boundary points are Wiener regular. Consider $L_0 u=-\mathrm{div}(A_0\nabla u)$, $Lu=-\mathrm{div}(A\nabla u)$, two real (non-necessarily symmetric) uniformly elliptic operators in $Ω$, and write $ω_{L_0}$, $ω_L$ for the respective associated elliptic measures. The goal of this program is to find sufficient conditions guaranteeing that $ω_L$ satisfies an $A_\infty$-condition or a $RH_q$-condition with respect to $ω_{L_0}$. In this paper we are interested in obtaining square function and non-tangential estimates for solutions of operators as before. We establish that bounded weak null-solutions satisfy Carleson measure estimates, with respect to the associated elliptic measure. We also show that for every weak null-solution, the associated square function can be controlled by the non-tangential maximal function in any Lebesgue space with respect to the associated elliptic measure. These results extend previous work of Dahlberg-Jerison-Kenig and are fundamental for the proof of the perturbation results in arXiv:1901.08261.
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