Let $n \in \mathbb{N}$. For each $m \in \{0,-1\}$, let $S^m_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ denote the class of smooth functions $b: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{C}$ such that, for every pair of multi-indices $\alpha,\beta \in \mathbb{N}_0^n$, the seminorm

\begin{align*} q_{m,\alpha,\beta}(b) := \sup_{(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n} (1+|\xi|)^{-m+|\beta|}|\partial_x^\alpha \partial_\xi^\beta b(x,\xi)| \end{align*}

is finite. Let $\mathcal{S}(\mathbb{R}^n)$ be the [Schwartz space](/page/Schwartz%20Space) and let $\mathcal{S}'(\mathbb{R}^n)$ be the space of [tempered distributions](/page/Tempered%20Distributions). For $b \in S^0_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$, let $\operatorname{Op}(b): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ denote the Kohn-Nirenberg quantisation of $b$, and let $\operatorname{Op}^w(b): \mathcal{S}(\mathbb{R}^n) \to \mathcal{S}'(\mathbb{R}^n)$ denote the Weyl quantisation of $b$. Let $a \in S^0_{1,0}(\mathbb{R}^n \times \mathbb{R}^n)$ be a real-valued symbol satisfying

\begin{align*} a(x,\xi) \geq 0 \end{align*}

for every $(x,\xi) \in \mathbb{R}^n \times \mathbb{R}^n$. Then there exists a constant $C \geq 0$, depending only on $n$ and finitely many seminorms $q_{0,\alpha,\beta}(a)$, such that

\begin{align*} \operatorname{Re}(\operatorname{Op}(a)u,u)_{L^2(\mathbb{R}^n)} \geq -C\|u\|_{L^2(\mathbb{R}^n)}^2 \end{align*}

for every $u \in \mathcal{S}(\mathbb{R}^n)$.