We are given (weak) Gårding's inequality for elliptic pseudodifferential operators:
Given $a\in S^m$ such that $\operatorname{Op}(a)$ is an elliptic operator, namely $\exists c,R>0$ such that for each $x,\xi\in\mathbb R^n$, we have $\operatorname{Re} a(x,\xi)\ge c(1+\lvert\xi\rvert)^m,\forall\lvert\xi\rvert\ge R$. Then for each $s\in\mathbb R$, there exists $A_s,B_s>0$ such that for each $u\in H^{m/2}$, we have $\operatorname{Re}\langle\operatorname{Op}(a)u,u\rangle\ge A_s\lVert u\rVert_{H^{m/2}}^2-B_s\lVert u\rVert_{H^s}^2$.
I wonder whether we can deduce the following version of Gårding-type inequality:
Suppose $\Omega\subseteq\mathbb R^n$ is a smooth bounded domain, and $L=\sum_{\lvert\alpha\rvert\le2k}a_\alpha\partial^\alpha$ is a real elliptic differential operator of order $2k$ on $\Omega$, namely the associated symbol $a(x,\xi)=\sum_\alpha a_\alpha(x)\xi^\alpha$ satisfies elliptic condition: there exists $c,R>0$ such that $a(x,\xi)\ge c(1+\lvert\xi\rvert)^{2k},\forall x\in\Omega,\lvert\xi\rvert\ge R$, and $a_\alpha\in C_b^\infty(\Omega,\mathbb R)$, the space of real smooth functions with all partial derivatives bounded. Then there exists $A_0,B_0>0$ such that for each $u\in H_0^k(\Omega)$, we have $\langle Lu,u\rangle\ge A_0\lVert u\rVert_{H^k}^2-B_0\lVert u\rVert_{L^2}^2$.
It seems to me that direct extension doesn't work, since we cannot extend $L$ to an elliptic operator on $\mathbb R^n$ (Whitney extension theorem needs compatible conditions near boundary). I wonder whether it's still a corollary of the inequality for that of $\mathbb R^n$ with a clever argument.
Any help is welcome.