Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for simplicity, assume $A \equiv$ Id) with fixed pole $p\in\Omega$ far from the boundary.
It is known (for example see theorem 1.8 in Gruter and Widman) that there exists $\alpha \in (0,1)$ depending on the Lipschitz constant of the boundary of $\Omega$ and the ellipticity of $A(x)$ such that $$ g(x) \lesssim \operatorname{dist}(x,\partial\Omega)^\alpha |x-p|^{2-d-\alpha}. $$ In particular, since I assume that $p$ is far from $x$ and the boundary, we have $$ g(x) \lesssim \operatorname{dist}(x,\partial\Omega)^\alpha. $$
Nonetheless, I have the feeling that this estimate is very pessimistic. It is obvious that this is the adequate growth near the vertex of a cone, for example. But I wonder if we have "better behavior" on average. In particular, fix a ball $B$ centered on $\partial\Omega$ and denote $\Sigma = B \cap \partial\Omega$. Let $\Sigma_\delta$ be the set of points in $\Omega$ such that its distance to $\Sigma$ is smaller than $\delta$. Is it true that $$ \frac{1}{|\Sigma_\delta|}\int_{\Sigma_\delta} g(x)\, dx \leq C\delta^2 \quad\mbox{as $\delta\to0$ ?} $$ My intuition is that, on average, $g(x)$ should behave as $\operatorname{dist}(0,\partial\Omega)$.