Let $\Omega$ be a sufficiently regular domain of $\mathbb{R}^2$ and let $h>0$ be the size of a triangulation of $\Omega$. Define $$ U_h:=\{x\in\Omega: \operatorname{dist}(x,\partial\Omega)\le h \}. $$ I would like to prove the following inequality: $$ |u|_{H^s(U_h)}\le C(h) h^{\frac{1}{2}} \|u \|_{H^{p+1}(\Omega)}\qquad\forall\ u\in H^{p+1}(\Omega), $$ for every $0\le s \le p+\frac{1}{2}-\varepsilon$, $\varepsilon>0$.
Any idea or comment would be really appreciated!