Let $M$ be a compact smooth $n$-manifold equipped with a smooth positive density $d\mu$, and let $h_0>0$. Let $(U,\kappa)$ be a coordinate chart with $\kappa(U)=V\subset\mathbb{R}^n$. Let $\chi,\rho\in C_c^\infty(V;\mathbb{R})$ satisfy $\rho=1$ on an open neighbourhood of $\operatorname{supp}\chi$.
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For $u\in C^\infty(M)$, define $T_\rho u\in C_c^\infty(\mathbb{R}^n)$ by
and by $T_\rho u(x)=0$ for $x\in\mathbb{R}^n\setminus V$.
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Let $b\in S^0(V\times\mathbb{R}^n;\mathbb{R})$ be a real semiclassical symbol, with its $S^0$ seminorms uniformly bounded for $0<h\le h_0$. Assume that there exists an open set $W\subset V$ with $\operatorname{supp}\chi\subset W$ such that
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\begin{align*}
b(x,\xi;h)\ge 0
\end{align*}
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for every $x\in W$, every $\xi\in\mathbb{R}^n$, and every $0<h\le h_0$.
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Let $q\in S^0(\mathbb{R}^n\times\mathbb{R}^n;\mathbb{R})$ be the zero extension of the symbol $\chi(x)^2 b(x,\xi;h)$ from $V\times\mathbb{R}^n$ to $\mathbb{R}^n\times\mathbb{R}^n$. Define the left semiclassical Kohn--Nirenberg quantization by
for $v\in C_c^\infty(\mathbb{R}^n)$, interpreted as the usual oscillatory integral.
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Then there exist constants $C>0$ and $h_1\in(0,h_0]$, depending only on finitely many $S^0$ seminorms of $b$ on a fixed neighbourhood of $\operatorname{supp}\chi\times\mathbb{R}^n$, finitely many $C^k$ seminorms of $\chi$ and $\rho$, and the fixed chart-density norm comparison, such that for every $u\in C^\infty(M)$ and every $0<h\le h_1$,
Equivalently, a localized nonnegative order-zero symbol contributes at worst an $O(h)\|u\|_{L^2(M,d\mu)}^2$ error in the corresponding microlocal energy estimate.