Green function estimates on complements of low-dimensional uniformly rectifiable sets

TL;DR

This paper establishes that for certain elliptic operators on domains with lower-dimensional uniformly rectifiable boundaries, the Green function closely approximates a power of the boundary distance function, with estimates linked to geometric properties.

Contribution

It extends previous weak Green function estimates to strong, more precise bounds for degenerate elliptic operators on lower-dimensional uniformly rectifiable sets.

Findings

Green function approximates boundary distance function raised to a power

Carleson measure estimate for the gradient of the Green function ratio

Results differ from previous weak estimates by using integration by parts and geometric properties

Abstract

It has been recently established by the first and third author 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 $L_{\beta,\gamma} =- {\rm div} D^{d+1+\gamma-n} \nabla$ associated to a domain $\Omega \subset \mathbb R^n$ with a uniformly rectifiable boundary $\Gamma$ of dimension $d < n-1$, the now usual distance to the boundary $D = D_\beta$ given by $D_\beta(X)^{-\beta} = \int_{\Gamma} |X-y|^{-d-\beta} d\sigma(y)$ for $X \in \Omega$, where $\beta >0$ and $\gamma \in (-1,1)$. In this paper we show that the Green function $G$ for $L_{\beta,\gamma}$, with pole at infinity, is well approximated by multiples of $D^{1-\gamma}$, in the sense that the function $\big| D\nabla\big(\ln\big( \frac{G}{D^{1-\gamma}} \big)\big)\big|^2$ satisfies a Carleson measure estimate on $\Omega$. 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 a previous work from the first author, the third author, and Max Engelstein.