David Guy, Feneuil Joseph, Mayboroda Svitlana
Laboratoire de mathématiques d'Orsay, Université Paris-Saclay, CNRS, 91405 Orsay, France.
School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA.
Math Ann. 2023;385(3-4):1797-1821. doi: 10.1007/s00208-022-02379-8. Epub 2022 Mar 12.
It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions. arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the "flagship" degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain with a uniformly rectifiable boundary of dimension , the now usual distance to the boundary given by for , where and . In this paper we show that the Green function for , with pole at infinity, is well approximated by multiples of , in the sense that the function satisfies a Carleson measure estimate on . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the "magical" distance function from David et al. (Duke Math J, to appear).
最近在大卫和马约罗达（《所有维度具有一致可求长边界的格林函数和区域的逼近。arXiv:2010.09793》）的研究中表明，在一致可求长集上，格林函数在弱意义下几乎是仿射的，而且，在某些情形下，这样的格林函数估计与集合的一致可求长性是等价的。本文处理这些结果的一个强类似情形，从具有低维边界的集合上的“旗舰”退化算子开始。我们考虑与一个具有维度为(n)的一致可求长边界(\partial\Omega)的区域(\Omega)相关联的椭圆算子，对于(x\in\Omega)，到边界的通常距离由(d(x,\partial\Omega)=\inf_{y\in\partial\Omega}|x - y|)给出，其中(n\geq2)且(\Omega\subset\mathbb{R}^n)。在本文中，我们表明对于(\Omega)，极点在无穷远处的格林函数(G(x,\infty))被(d(x,\partial\Omega))的倍数很好地逼近，即函数(\frac{G(x,\infty)}{d(x,\partial\Omega)})在(\Omega)上满足卡尔松测度估计。我们强调强结果和弱结果在本质上是不同的，当然，在证明层面也是如此：后者广泛使用紧致性论证，而本文依赖于一些复杂的分部积分以及来自大卫等人（《杜克数学杂志》，即将发表）的“神奇”距离函数的性质。
