 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 $\lambda \ge 0$ and $0<\theta <1$, there is ${\delta }_0={\delta }_0(n,\lambda ,\theta )>0$ such that, if $0<\delta \le {\delta }_0$ and every admissible nested pair satisfiesthenThe 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.