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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.

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(This should be a comment, but I don't have enough reputation).

The related question Criteria for Positivity of Pseudoddifferential Operators on Manifolds and the paper of Fefferman and Phong mentioned there seems to shed some light on some of your questions.

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  • $\begingroup$ Thank you. I noticed the post and the paper before, but expected getting some quick answer. Now it looks like I can't avoid reading it myself. $\endgroup$ – Liren Lin Dec 3 '15 at 9:31
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The Wick quantization has the property you are interested in. This article by N. Lerner about the Wick pseudodifferential calculus may be of interest to you.

Concerning the earlier post mentioned by Dylan, please be warned that the 1982 paper by Fefferman and Phong, which was refered to in the accepted answer to that post, defers the proof of a crucial Geometrical Lemma to a more complete version of their article. Perhaps the completion is contained in the (very long) 1983 paper The Uncertainty Principle by Fefferman, but I am not sure.

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