  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.
