Let $S$ be the space of *symbols* defined by $$S:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}\},$$ and recall that the *Weyl quantization* of a symbol is given by $$(\mathrm{Op}_h(a)u)(x):=\frac{1}{(2\pi h)^d}\iint e^{\frac{i}{h}\langle x-y,\xi\rangle}a\left(\frac{x+y}{2},\xi\right)u(y)\,dy\,d\xi.$$
The *Calderon-Vaillancourt theorem* says that $\mathrm{Op}_h(a)$ is bounded on $L^2(\mathbb{R}^d)$ uniformly in $h$, with bound given by $$\|\mathrm{Op}_h(a)\|_{L^2\to L^2}\le C\sum_{|\alpha|\le Kd}h^{\frac{|\alpha|}{2}}\|\partial^{\alpha}a\|_{L^{\infty}}.$$ In particular, this implies a uniform bound even when $a=a_h$ is $h$-dependent, as long as it is in the class $$S_{\delta}:=\{a\in C^{\infty}(T^*\mathbb{R}^d):\forall \alpha,\beta\in\mathbb{Z}^d,\, |\partial_x^{\alpha}\partial_{\xi}^{\beta}a(x,\xi)|\le C_{\alpha\beta}h^{-\delta|\alpha+\beta|}\}$$ with $\delta\le\frac{1}{2}$. Specifically, if $a\in S_{\delta}$, then $$\|\mathrm{Op}_h(a)\|_{L^2\to L^2}\le\|a\|_{L^{\infty}}+O(h^{½-\delta}).$$

All of the above is standard, and spelled out in the books by Zworski, Martinez, and Dimassi–Sjöstrand. But in Chapter 13 of Zworski, we learn that it is not the full story. In fact (Theorem 13.13 in Zworski), it holds that for $a\in S$, $$\|\mathrm{Op}_h(a)\|_{L^2\to L^2}\le \|a\|_{L^{\infty}}+O(h).$$ The proof is fairly complicated, using the FBI transform. **My questions are of the derivative dependence for this result.** What powers of $h$ accompany which derivatives of $a$? In particular, **can the error bound for $S_{\delta}$ symbols be strengthened to something better than $O(h^{½-\delta})$?** And whatever the optimal bound is, **has it ever been shown to be sharp?**