[proofplan] We first insert the cutoffs and observe that the localized kernel is smooth, compactly supported, and rapidly small with all derivatives. We then prove a classical Sobolev estimate for compactly supported smooth-kernel operators by moving derivatives onto the kernel and applying Cauchy-Schwarz on a fixed compact set. Finally, we compare classical and semiclassical Sobolev norms; the comparison may cost a fixed power of $h^{-1}$, but the kernel is $O(h^M)$ for every $M$, so we choose $M$ large enough to absorb that loss and obtain the desired $O(h^N)$ estimate. [/proofplan]

[step:Localize the kernel and record its rapid derivative bounds] Define the localized kernel \begin{align*} K_{\chi,\psi}:\mathbb{R}^n\times\mathbb{R}^n\times(0,h_0]\to\mathbb{C} \end{align*} by \begin{align*} K_{\chi,\psi}(x,y;h)=\chi(x)K_R(x,y;h)\psi(y). \end{align*} Let $X=\operatorname{supp}\chi\subset\mathbb{R}^n$ and $Y=\operatorname{supp}\psi\subset\mathbb{R}^n$. Then $X$ and $Y$ are compact, and $K_{\chi,\psi}(\cdot,\cdot;h)$ is supported in $X\times Y$. For every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$ and every $M\in\mathbb{N}$, Leibniz's rule and the local rapid-smoothing hypothesis on the compact set $X\times Y$ give a constant $C_{\alpha,\beta,M,\chi,\psi}>0$ such that \begin{align*} \sup_{(x,y)\in\mathbb{R}^n\times\mathbb{R}^n}\left|\partial_x^\alpha\partial_y^\beta K_{\chi,\psi}(x,y;h)\right|\le C_{\alpha,\beta,M,\chi,\psi}h^M \end{align*} for all $0<h\le h_0$. The operator $\chi R_h\psi$ is the integral operator with kernel $K_{\chi,\psi}$: \begin{align*} (\chi R_h\psi u)(x)=\int_{\mathbb{R}^n}K_{\chi,\psi}(x,y;h)u(y)\,d\mathcal{L}^n(y). \end{align*} [/step]

[step:Estimate the localized operator between classical Sobolev spaces by duality]For an integer $m\in\mathbb{N}_0$, let $H^m(\mathbb{R}^n)$ denote the classical $L^2$ Sobolev space with norm obtained by summing the $L^2(\mathbb{R}^n,\mathcal{L}^n)$ norms of $D^\alpha v$ over $|\alpha|\le m$. Let $H^{-m}(\mathbb{R}^n)$ denote the Hilbert-space dual of $H^m(\mathbb{R}^n)$ under the $L^2$ pairing. Choose integers $a,b\in\mathbb{N}_0$ such that $a\ge t$ and $b\ge -s$. We prove that, for every $M\in\mathbb{N}$, there exists $C_{a,b,M,\chi,\psi}>0$ such that \begin{align*} \|\chi R_h\psi u\|_{H^a(\mathbb{R}^n)}\le C_{a,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ and all $0<h\le h_0$. Fix $\gamma\in\mathbb{N}_0^n$ with $|\gamma|\le a$ and fix $\varphi\in C_c^\infty(\mathbb{R}^n)$. Since the $x$-support of $K_{\chi,\psi}$ is contained in $X$, Fubini's theorem applies to the compactly supported smooth integrand and gives \begin{align*} \int_{\mathbb{R}^n}D_x^\gamma(\chi R_h\psi u)(x)\overline{\varphi(x)}\,d\mathcal{L}^n(x)=\int_{\mathbb{R}^n}u(y)\overline{G_{\gamma,\varphi}(y;h)}\,d\mathcal{L}^n(y), \end{align*} where $G_{\gamma,\varphi}(\cdot;h):\mathbb{R}^n\to\mathbb{C}$ is defined by \begin{align*} G_{\gamma,\varphi}(y;h)=\int_X \overline{D_x^\gamma K_{\chi,\psi}(x,y;h)}\varphi(x)\,d\mathcal{L}^n(x). \end{align*} The function $G_{\gamma,\varphi}(\cdot;h)$ is smooth and supported in $Y$. Let $\delta\in\mathbb{N}_0^n$ satisfy $|\delta|\le b$. Differentiating under the integral sign in $y$ gives \begin{align*} D_y^\delta G_{\gamma,\varphi}(y;h)=\int_X \overline{D_y^\delta D_x^\gamma K_{\chi,\psi}(x,y;h)}\varphi(x)\,d\mathcal{L}^n(x). \end{align*} Applying the Cauchy-Schwarz inequality on $(X,\mathcal{B}(X),\mathcal{L}^n)$ and then using the derivative bound from the previous step gives \begin{align*} |D_y^\delta G_{\gamma,\varphi}(y;h)|\le \mathcal{L}^n(X)^{1/2}C_{\gamma,\delta,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Since $D_y^\delta G_{\gamma,\varphi}(\cdot;h)$ is supported in $Y$, integration over $Y$ yields \begin{align*} \|D_y^\delta G_{\gamma,\varphi}(\cdot;h)\|_{L^2(\mathbb{R}^n)}\le \mathcal{L}^n(X)^{1/2}\mathcal{L}^n(Y)^{1/2}C_{\gamma,\delta,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Summing over $|\delta|\le b$, we obtain \begin{align*} \|G_{\gamma,\varphi}(\cdot;h)\|_{H^b(\mathbb{R}^n)}\le C_{\gamma,b,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} By the definition of the negative Sobolev norm as the dual norm of $H^b(\mathbb{R}^n)$, \begin{align*} \left|\int_{\mathbb{R}^n}u(y)\overline{G_{\gamma,\varphi}(y;h)}\,d\mathcal{L}^n(y)\right|\le \|u\|_{H^{-b}(\mathbb{R}^n)}\|G_{\gamma,\varphi}(\cdot;h)\|_{H^b(\mathbb{R}^n)}. \end{align*} Therefore \begin{align*} \left|\int_{\mathbb{R}^n}D_x^\gamma(\chi R_h\psi u)(x)\overline{\varphi(x)}\,d\mathcal{L}^n(x)\right|\le C_{\gamma,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)}\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Taking the supremum over all $\varphi\in C_c^\infty(\mathbb{R}^n)$ with $\|\varphi\|_{L^2(\mathbb{R}^n)}\le 1$ gives \begin{align*} \|D_x^\gamma(\chi R_h\psi u)\|_{L^2(\mathbb{R}^n)}\le C_{\gamma,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)}. \end{align*} Summing over all $\gamma$ with $|\gamma|\le a$ proves the asserted classical Sobolev estimate.[/step]

[guided]The input space is allowed to have negative Sobolev order, so we cannot bound the local $L^2$ norm of $u$ by $\|u\|_{H^{-b}}$. Instead, we test the output against an $L^2$ function and move the required input regularity onto the kernel through derivatives in the $y$ variable. Fix $\gamma\in\mathbb{N}_0^n$ with $|\gamma|\le a$ and fix a test function $\varphi\in C_c^\infty(\mathbb{R}^n)$. Since $K_{\chi,\psi}$ is smooth and supported in $X\times Y$, the integrand below is smooth and compactly supported in $X\times Y$. Fubini's theorem therefore applies and gives \begin{align*} \int_{\mathbb{R}^n}D_x^\gamma(\chi R_h\psi u)(x)\overline{\varphi(x)}\,d\mathcal{L}^n(x)=\int_{\mathbb{R}^n}u(y)\overline{G_{\gamma,\varphi}(y;h)}\,d\mathcal{L}^n(y), \end{align*} where the function $G_{\gamma,\varphi}(\cdot;h):\mathbb{R}^n\to\mathbb{C}$ is defined by \begin{align*} G_{\gamma,\varphi}(y;h)=\int_X \overline{D_x^\gamma K_{\chi,\psi}(x,y;h)}\varphi(x)\,d\mathcal{L}^n(x). \end{align*} This function is smooth because the kernel is smooth, and it is supported in $Y$ because $K_{\chi,\psi}(x,y;h)=0$ for $y\notin Y$. To pair $u\in H^{-b}(\mathbb{R}^n)$ with $G_{\gamma,\varphi}$, we must prove that $G_{\gamma,\varphi}\in H^b(\mathbb{R}^n)$ with a rapidly small norm. Let $\delta\in\mathbb{N}_0^n$ satisfy $|\delta|\le b$. Differentiating under the integral sign in $y$ is justified by smoothness and compact support, and yields \begin{align*} D_y^\delta G_{\gamma,\varphi}(y;h)=\int_X \overline{D_y^\delta D_x^\gamma K_{\chi,\psi}(x,y;h)}\varphi(x)\,d\mathcal{L}^n(x). \end{align*} Apply the Cauchy-Schwarz inequality on the measure space $(X,\mathcal{B}(X),\mathcal{L}^n)$ to the two functions $x\mapsto D_y^\delta D_x^\gamma K_{\chi,\psi}(x,y;h)$ and $x\mapsto\varphi(x)$. The rapid derivative bound for the localized kernel gives \begin{align*} |D_y^\delta G_{\gamma,\varphi}(y;h)|\le \mathcal{L}^n(X)^{1/2}C_{\gamma,\delta,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Because $D_y^\delta G_{\gamma,\varphi}$ is supported in $Y$, integrating this pointwise estimate over $Y$ with respect to $\mathcal{L}^n$ gives \begin{align*} \|D_y^\delta G_{\gamma,\varphi}(\cdot;h)\|_{L^2(\mathbb{R}^n)}\le \mathcal{L}^n(X)^{1/2}\mathcal{L}^n(Y)^{1/2}C_{\gamma,\delta,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Summing this estimate over all $\delta$ with $|\delta|\le b$ proves \begin{align*} \|G_{\gamma,\varphi}(\cdot;h)\|_{H^b(\mathbb{R}^n)}\le C_{\gamma,b,M,\chi,\psi}h^M\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Now the definition of $H^{-b}(\mathbb{R}^n)$ as the dual of $H^b(\mathbb{R}^n)$ gives \begin{align*} \left|\int_{\mathbb{R}^n}u(y)\overline{G_{\gamma,\varphi}(y;h)}\,d\mathcal{L}^n(y)\right|\le \|u\|_{H^{-b}(\mathbb{R}^n)}\|G_{\gamma,\varphi}(\cdot;h)\|_{H^b(\mathbb{R}^n)}. \end{align*} Combining the preceding estimates, we obtain \begin{align*} \left|\int_{\mathbb{R}^n}D_x^\gamma(\chi R_h\psi u)(x)\overline{\varphi(x)}\,d\mathcal{L}^n(x)\right|\le C_{\gamma,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)}\|\varphi\|_{L^2(\mathbb{R}^n)}. \end{align*} Taking the supremum over all $\varphi\in C_c^\infty(\mathbb{R}^n)$ with $\|\varphi\|_{L^2(\mathbb{R}^n)}\le 1$ gives \begin{align*} \|D_x^\gamma(\chi R_h\psi u)\|_{L^2(\mathbb{R}^n)}\le C_{\gamma,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)}. \end{align*} Finally, summing over $|\gamma|\le a$ proves \begin{align*} \|\chi R_h\psi u\|_{H^a(\mathbb{R}^n)}\le C_{a,b,M,\chi,\psi}h^M\|u\|_{H^{-b}(\mathbb{R}^n)}. \end{align*}[/guided]

[step:Compare classical and semiclassical Sobolev norms] For $r\in\mathbb{R}$, the space $H_h^r(\mathbb{R}^n)$ and the Fourier transform notation $\widehat{v}$ are as defined in the theorem statement. Since $a\ge t$ and $0<h\le h_0$, there exists $C_{a,t,h_0}>0$ such that \begin{align*} \|v\|_{H_h^t(\mathbb{R}^n)}\le C_{a,t,h_0}\|v\|_{H^a(\mathbb{R}^n)} \end{align*} for all $v\in H^a(\mathbb{R}^n)$. Similarly, since $b\ge -s$, there exist constants $A=A(s,b)\ge 0$ and $C_{s,b,h_0}>0$ such that \begin{align*} \|w\|_{H^{-b}(\mathbb{R}^n)}\le C_{s,b,h_0}h^{-A}\|w\|_{H_h^s(\mathbb{R}^n)} \end{align*} for all $w\in H_h^s(\mathbb{R}^n)$ and all $0<h\le h_0$. This follows by taking the supremum over $\xi\in\mathbb{R}^n$ of the ratio \begin{align*} \frac{(1+|\xi|^2)^{-b/2}}{(1+h^2|\xi|^2)^{s/2}}, \end{align*} which is bounded by a fixed power of $h^{-1}$ because $b+s\ge 0$. [/step]

[step:Absorb the fixed semiclassical loss into rapid kernel decay] Let $N\in\mathbb{N}$ be prescribed. Choose $M\in\mathbb{N}$ with $M\ge N+A$, where $A=A(s,b)$ is the exponent from the preceding norm comparison. Combining the classical estimate with the two norm comparisons gives \begin{align*} \|\chi R_h\psi u\|_{H_h^t(\mathbb{R}^n)}\le C_{s,t,M,\chi,\psi}h^{M-A}\|u\|_{H_h^s(\mathbb{R}^n)}. \end{align*} Since $M-A\ge N$, and since $0<h\le h_0$, we may absorb the harmless factor $h_0^{M-A-N}$ into the constant. Hence there exists $C_{s,t,N,\chi,\psi}>0$ such that \begin{align*} \|\chi R_h\psi u\|_{H_h^t(\mathbb{R}^n)}\le C_{s,t,N,\chi,\psi}h^N\|u\|_{H_h^s(\mathbb{R}^n)}. \end{align*} The estimate was first obtained for smooth compactly supported $u$. By density of $C_c^\infty(\mathbb{R}^n)$ in $H_h^s(\mathbb{R}^n)$ for finite $s$, the operator extends uniquely to a bounded map \begin{align*} \chi R_h\psi:H_h^s(\mathbb{R}^n)\to H_h^t(\mathbb{R}^n) \end{align*} with the same bound. This proves \begin{align*} \|\chi R_h\psi\|_{\mathcal{L}(H_h^s(\mathbb{R}^n),H_h^t(\mathbb{R}^n))}=O(h^N) \end{align*} for every $N\in\mathbb{N}$, completing the proof. [/step]