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.
