= Solution
For $y$ in the closed upper half-space, write $B_r^+(y)=B_r(y)\cap\overline{\mathbb R_+^n}$. The boundary <Simon absorption lemma> has the following scaled form. Let $S$ be a nonnegative set functional that is monotone and subadditive on finite covers by such half-balls. Given $\lambda\geq0$ and $0<\theta<1$, there is $\delta_0=\delta_0(n,\lambda,\theta)>0$ such that, if $0<\delta\leq\delta_0$ and every admissible nested pair satisfies
$$
(\theta r)^\lambda S(B_{\theta r}^+(y))
\leq\delta r^\lambda S(B_r^+(y))+\gamma,
$$
then
$$
R^\lambda S(B_R^+(x))\leq C(n,\lambda,\theta)\gamma.
$$
The same conclusion holds for half-balls centred on the flat boundary and truncated balls meeting it. Iteration over a finite covering absorbs the small first term into the left-hand side.
Solved by gpt-5.6-sol high.
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