Easy Garding Inequality states that if $a=a(x,\xi)$ is a symbol in $S=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha} \hspace{2mm} \forall \alpha\}$ with $a\geq \gamma >0 $ on $\mathbb{R}^{2n}$ then for $h$ sufficiently small we have that: $$ \langle a^{w}(x,hD)u,u \rangle \geq (\gamma-\epsilon) \|u\|_{L^2}^2$$

My question is now precisely about what happens when the symbol $a$ does not belong to $S$ but rather $ a\in S(m)=\{a\in C^{\infty}||\partial_{\alpha}a|<C_{\alpha}m \hspace{2mm} \forall \alpha\}$ for some order function $m$ (For example $m(x) = \langle x \rangle^k$ ) Can we write the easy Garding Inequality with with the $L^2$ norms replaced by weighted Sobolev norms? Thanks