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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

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I understand that you are dealing with semi-classical symbols $$ b(x,\xi, h)=a(x,h\xi), \quad \vert\partial_x^\alpha\partial_\xi^\beta b\vert\le C_{\alpha\beta} h^{\vert \beta\vert} m(x),\quad 0<h\le 1. $$ According to Hörmander's terminology, we have $$ b\in S(m, g),\quad g_{x,\xi}(t,\tau)={\vert dt\vert^2} +h^2{\vert d\tau\vert^2}. $$ If $b\ge 0$, then the standard Garding inequality says that there exists $c\in S(mh, g)$ such that $$ b^w(x,D)+c^w(x,D)\ge 0. $$ The Fefferman-Phong inequality, a drastic improvement of the latter, says that there exists $c\in S(mh^2, g)$ such that $$ b^w(x,D)+c^w(x,D)\ge 0. $$ Note that tou have to verify some mild assumption on the function $m$ for the above statements to hold true, e.g. $m(x)>0$ and such that $$\exists C,N,\forall x,y,\quad \frac{m(x+y)}{m(x)}\le C(1+\vert y\vert)^{N}, $$ and this is satisfied by your example.

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