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

Abstract

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 $L_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$ associated to a domain $\Omega \subset R^n$ with a uniformly rectifiable boundary $\Gamma$ of dimension $d<n-1$ , the now usual distance to the boundary $D=D_{\beta }$ given by $D_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$ 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 $|D\nabla (ln(\frac{G}{D^{1-\gamma }})){|}^2$ 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 David et al. (Duke Math J, to appear).