Let $ u \in C^{\infty}_{c}(\Omega) $ where $ \Omega $ is an bounded open set in $\mathbb{R}^{n}$ with smooth boundary. If $ 0 \leq h \leq 1 $ is it possible for us to establish the bound $$4h^2\|h\partial_{x_n}u\|_{L^2}^2 + 4h^2 \|u + hx_nu\|_{L^2}^2 \geq Ch^k \|u\|_{L^2}^2$$
where $ k =1 $. If not, what about the case where $ k \in \mathbb{N} $? Here $ x_n $ is the nth coordinate of $x \in \mathbb{R}^n$. The first step I came up with is to look the bound $$ 4h^2\|h\partial_{x_n}u\|_{L^2}^2 + 4h^2 \|u + hx_n u\|_{L^2}^2 \geq 4h^2| \langle h\partial_{x_n}u, u + hx_n u\rangle | $$ using the Cauchy Schwartz inequality. This bound appear in the proof of a Carleman estimate I have been trying to complete.
In the first equation the left hand side has terms of orders $h^2$, $h^3$ and $h^4$. The right hand side has $h^k$.
Let $h \to 0$. In the limit, the left hand side is or order $h^2$ and, due to the inequality, the right hand side must be of order $h^2$ or something that goes to zero faster. That is, $k \ge 2$.