Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)

Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\to\mathbb{C}$ be a symbol in $S_{2n}(\langle\xi\rangle^{m})$. That is, regarded as a function on $\mathbb{R}^{2n}$, $p$ is smooth, and for every multiindex $\alpha=(\alpha_1,\ldots,\alpha_{2n})$, there is a constant $C_\alpha$ such that $|\partial^\alpha p|\le C_\alpha\langle\xi\rangle^m$. Here $\langle\xi\rangle:=(1+|\xi|^2)^{1/2}$. Associate to such $p$ and small parameters $h>0$ we define the (Weyl) semiclassical pseudodifferential operator ($h$-PDO) $$\mathrm{Op}_h^W(p)u\,(x)=\frac{1}{(2\pi h)^n}\int e^{i(x-y)\cdot\xi/h}p(\frac{x+y}{2},\xi)u(y)\,dy\,d\xi\quad(u\in C^\infty_c(\mathbb{R}^n)).$$ I'm concerned with the positivity of $\mathrm{Op}_h^W(p)$, that is the positivity of $\langle\mathrm{Op}_h^W(p) u,u\rangle_{L^2}$. We have the following result.

Sharp Gårding inequality If $p\ge 0$, then there exists a constant $C>0$ such that $$\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch\|u\|_{H^{m/2}}^2\tag{1}$$ for all small enough $h>0$ (i.e. for all $h$ smaller than some positive constant depending on $C$) and for all $u\in C^\infty_c(\mathbb{R}^n)$.

When $m=0$ (in fact, I don't know if this is needed), the Fefferman-Phong Inequality improves the result to $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch^2\|u\|_{L^2}^2$. However, positivity of $\mathrm{Op}_h^W(p)$ is still not guaranteed.

On the other hand, the Easy Gårding inequality replaces the assumption $p\ge 0$ by $p\ge p_0\langle\xi\rangle^m$ for some constant $p_0>0$, and the conclusion is $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge C\|u\|_{H^{m/2}}^2$ for every $0<C<p_0$ (the larger $C$ is chosen, the smaller $h$ is required). Thus, positivity of $\mathrm{Op}_h^W(p)$ is obtained.

Question. If $p\ge 0$, and there exist constants $p_0>0$ and $R>0$ such that $p\ge p_0\langle\xi\rangle^m$ for $|\xi|\ge R$, is it possible to add some "smallness" assumption on the size of the zero set of $p$ to guarantee $\mathrm{Op}_h^W(p)\ge 0$?

The model example in my mind is $p\in |\xi|^2\in S_{2n}(\langle\xi\rangle^2)$. In this case $p=0$ only at $\xi =0$ (precisely, on $\mathbb{R}^n_x\times\{\xi=0\}$), while  $\mathrm{Op}_h^W(|\xi|^2) = -h^2\Delta\ge 0$, since $$\langle-h^2\Delta u,u\rangle_{L^2} = h^2\int |\nabla u|^2.$$ Even for this simple example, I don't know how to obtain the positivity in the framework of $h$-PDO theory. (I'm not very familiar with the theory of $h$-PDO though.)

Is it true that $\mathrm{Op}_h^W(p)\ge 0$ if, besides the assumptions given in Question, $p=0$ only at a single $\xi$? How about $p=0$ on a small set in $\mathbb{R}^n_x\times\{|\xi|<R\}$, say with small Hausdorff dimension? If yes, are there generalizations to systems (matrix-valued $p$)? Or there are counterexamples for these questions?

Indeed, the case of matrix-valued $p$ is crucial for my current research. But any partial answer or relevant references will be appreciated.

Let me first give some background. (My reference is Martinez's book An introduction to semiclassical and microlocal analysis)

Let $m\in\mathbb{R}$, and $p(x,\xi):\mathbb{R}^n_x\times\mathbb{R}^n_\xi\to\mathbb{C}$ be a symbol in $S_{2n}(\langle\xi\rangle^{m})$. That is, regarded as a function on $\mathbb{R}^{2n}$, $p$ is smooth, and for every multiindex $\alpha=(\alpha_1,\ldots,\alpha_{2n})$, there is a constant $C_\alpha$ such that $|\partial^\alpha p|\le C_\alpha\langle\xi\rangle^m$. Here $\langle\xi\rangle:=(1+|\xi|^2)^{1/2}$. Associate to such $p$ and small parameters $h>0$ we define the (Weyl) semiclassical pseudodifferential operator ($h$-PDO) $$\mathrm{Op}_h^W(p)u\,(x)=\frac{1}{(2\pi h)^n}\int e^{i(x-y)\cdot\xi/h}p(\frac{x+y}{2},\xi)u(y)\,dy\,d\xi\quad(u\in C^\infty_c(\mathbb{R}^n)).$$ I'm concerned with the validity of $\mathrm{Op}_h^W(p)\ge 0$, that is $\langle\mathrm{Op}_h^W(p) u,u\rangle_{L^2}\ge 0$ for all $u\in C^\infty_c(\mathbb{R}^n)$. The following result is well-known.

Sharp Gårding inequality If $p\ge 0$, then there exists a constant $C>0$ such that $$\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch\|u\|_{H^{m/2}}^2\tag{1}$$ for all small enough $h>0$ (i.e. for all $h$ smaller than some positive constant depending on $C$) and for all $u\in C^\infty_c(\mathbb{R}^n)$.

• When $m=0$ (in fact, I don't know if this is needed), the Fefferman-Phong Inequality improves the result to $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge -Ch^2\|u\|_{L^2}^2$. However, $\mathrm{Op}_h^W(p)\ge 0$ is still not guaranteed.
• On the other hand, the Easy Gårding inequality replaces the assumption $p\ge 0$ by $p\ge p_0\langle\xi\rangle^m$ for some constant $p_0>0$, and the conclusion is $\langle\mathrm{Op}_h^W(p)u,u\rangle_{L^2}\ge C\|u\|_{H^{m/2}}^2$ for every $0<C<p_0$ (the larger $C$ is chosen, the smaller $h$ is required). Thus, $\mathrm{Op}_h^W(p)\ge 0$.

Question. If $p\ge 0$, and there exist constants $p_0>0$ and $R>0$ such that $p\ge p_0\langle\xi\rangle^m$ for $|\xi|\ge R$, is it possible to add some "smallness" assumption on the size of the zero set of $p$ to guarantee $\mathrm{Op}_h^W(p)\ge 0$?

The model example in my mind is $p\in |\xi|^2\in S_{2n}(\langle\xi\rangle^2)$. In this case $p=0$ only at $\xi =0$ (precisely, on $\mathbb{R}^n_x\times\{\xi=0\}$). We cannot apply the Easy Gårding inequality, while $\mathrm{Op}_h^W(|\xi|^2) = -h^2\Delta\ge 0$, since $$\langle-h^2\Delta u,u\rangle_{L^2} = h^2\int |\nabla u|^2.$$

Here are some more concrete questions:

• Is it true that $\mathrm{Op}_h^W(p)\ge 0$ if, besides the assumptions given in Question, $p=0$ only at a single $\xi$?
• How about $p=0$ on a small set in $\mathbb{R}^n_x\times\{|\xi|<R\}$, say with small Hausdorff dimension?
• If yes, are there generalizations to systems (matrix-valued $p$)?
• Or there are counterexamples for these questions?

Indeed, the case of matrix-valued $p$ is crucial for my current research. But any partial answer or relevant references will be appreciated.