## Formalized Name Residual Operator Criterion ## Formalized Statement Fix $n\in\mathbb{N}$ and $h_0>0$. For $r\in\mathbb{R}$, let $H_h^r(\mathbb{R}^n)$ denote the semiclassical Sobolev space with norm \begin{align*} \|u\|_{H_h^r}=\|\langle hD\rangle^r u\|_{L^2(\mathbb{R}^n)} \end{align*} where $\langle hD\rangle^r$ is the Fourier multiplier with symbol $\langle h\xi\rangle^r=(1+h^2|\xi|^2)^{r/2}$. Let $(R_h)_{0<h\leq h_0}$ be a family of continuous operators $C_c^\infty(\mathbb{R}^n)\to C^\infty(\mathbb{R}^n)$ with smooth Schwartz kernels $K_h:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$. Assume that for every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$ and every $L,N\in\mathbb{N}$ there is a constant $C_{\alpha,\beta,L,N}>0$ such that \begin{align*} \sup_{x,y\in\mathbb{R}^n}(1+|x-y|)^L|\partial_x^\alpha\partial_y^\beta K_h(x,y)|\leq C_{\alpha,\beta,L,N}h^N \end{align*} for all $0<h\leq h_0$. Then for every $s,t\in\mathbb{R}$ and every $N\in\mathbb{N}$ there is a constant $C_{s,t,N}>0$ such that \begin{align*} \|R_h u\|_{H_h^t}\leq C_{s,t,N}h^N\|u\|_{H_h^{-s}} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ and all $0<h\leq h_0$. Conversely, suppose that for every $\chi,\psi\in C_c^\infty(\mathbb{R}^n)$, every $s,t\in\mathbb{R}$, and every $N\in\mathbb{N}$, the localized family $\chi R_h\psi$ satisfies \begin{align*} \|\chi R_h\psi u\|_{H_h^t}\leq C_{\chi,\psi,s,t,N}h^N\|u\|_{H_h^{-s}} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ and all $0<h\leq h_0$. Then for every compact set $E\subset\mathbb{R}^n\times\mathbb{R}^n$, every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$, and every $N\in\mathbb{N}$ there is a constant $C_{E,\alpha,\beta,N}>0$ such that \begin{align*} \sup_{(x,y)\in E}|\partial_x^\alpha\partial_y^\beta K_h(x,y)|\leq C_{E,\alpha,\beta,N}h^N \end{align*} for all $0<h\leq h_0$. ## Proof [proofplan] The forward implication is a Schur-test estimate after differentiating the kernel enough times to represent the semiclassical Sobolev norms. The rapid weighted bounds make every differentiated kernel integrable in the difference variable $x-y$, and the arbitrary power of $h$ absorbs the finitely many semiclassical derivative factors. For the converse, compact cutoffs reduce the assertion to a compactly supported kernel; applying the localized operator to derivatives of point masses and then using semiclassical Sobolev embedding recovers pointwise kernel-derivative bounds, with the loss in powers of $h$ absorbed by the hypothesis that the mapping estimate holds for every $N$. [/proofplan] [step:Deduce rapid $L^2$ bounds for differentiated kernels] Let $\alpha,\beta\in\mathbb{N}_0^n$ and let $L>n$. Define the differentiated kernel $K_{h,\alpha,\beta}:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$ by \begin{align*} K_{h,\alpha,\beta}(x,y)=\partial_x^\alpha(-\partial_y)^\beta K_h(x,y). \end{align*} The sign appears because derivatives falling on the input are transferred to the $y$ variable by integration by parts. The hypothesis gives, for every $N\in\mathbb{N}$, \begin{align*} |K_{h,\alpha,\beta}(x,y)|\leq C_{\alpha,\beta,L,N}h^N(1+|x-y|)^{-L}. \end{align*} Since $L>n$, the function $z\mapsto(1+|z|)^{-L}$ belongs to $L^1(\mathbb{R}^n,d\mathcal{L}^n(z))$. Hence \begin{align*} \sup_{x\in\mathbb{R}^n}\int_{\mathbb{R}^n}|K_{h,\alpha,\beta}(x,y)|\,d\mathcal{L}^n(y)\leq C_{\alpha,\beta,L,N}h^N\int_{\mathbb{R}^n}(1+|z|)^{-L}\,d\mathcal{L}^n(z) \end{align*} and the same bound holds with the supremum over $y$ and integration in $x$. Schur's test therefore gives \begin{align*} \|\partial_x^\alpha R_h\partial_y^\beta u\|_{L^2(\mathbb{R}^n)}\leq C_{\alpha,\beta,N}'h^N\|u\|_{L^2(\mathbb{R}^n)} \end{align*} for all $u\in C_c^\infty(\mathbb{R}^n)$ after increasing the constant. [guided] The point of the weight $(1+|x-y|)^L$ is exactly to make the kernel integrable in either variable. Fix multi-indices $\alpha$ and $\beta$, and define \begin{align*} K_{h,\alpha,\beta}(x,y)=\partial_x^\alpha(-\partial_y)^\beta K_h(x,y). \end{align*} If $L>n$, then the comparison function $z\mapsto(1+|z|)^{-L}$ is integrable with respect to $n$-dimensional Lebesgue measure. The assumed residual estimate gives \begin{align*} |K_{h,\alpha,\beta}(x,y)|\leq C_{\alpha,\beta,L,N}h^N(1+|x-y|)^{-L}. \end{align*} Integrating in $y$ and using the translation $z=x-y$, whose Jacobian is one for Lebesgue measure, gives \begin{align*} \int_{\mathbb{R}^n}|K_{h,\alpha,\beta}(x,y)|\,d\mathcal{L}^n(y)\leq C_{\alpha,\beta,L,N}h^N\int_{\mathbb{R}^n}(1+|z|)^{-L}\,d\mathcal{L}^n(z). \end{align*} The right hand side is independent of $x$. The identical argument with the variables reversed gives the corresponding supremum in $y$. Schur's test applies to this integral kernel and yields an $L^2$ operator norm bounded by a constant times $h^N$. This proves the claimed rapid $L^2$ bound for every fixed pair of derivatives. [/guided] [/step] [step:Convert differentiated $L^2$ bounds into Sobolev mapping estimates] First suppose that $s,t\in\mathbb{N}_0$. The norm $\|R_hu\|_{H_h^t}$ is equivalent, with constants depending only on $n,t,h_0$, to the finite sum of $L^2$ norms of $(h\partial_x)^\alpha R_hu$ for $|\alpha|\leq t$. Similarly, the norm $\|u\|_{H_h^{-s}}$ is dual to the finite family of $H_h^s$ test norms. Applying the estimate from the previous step to all $|\alpha|\leq t$ and all $|\beta|\leq s$ gives \begin{align*} \|R_hu\|_{H_h^t}\leq C_{s,t,N}h^N\|u\|_{H_h^{-s}}. \end{align*} The extra factors $h^{|\alpha|}$ and $h^{|\beta|}$ from the semiclassical derivatives only improve the estimate for $0<h\leq h_0$, and any fixed loss is absorbed by replacing $N$ with a larger exponent in the residual kernel hypothesis. For arbitrary real $s,t$, choose integers $s_0,t_0$ with $s_0>|s|+1$ and $t_0>|t|+1$. The integer-order estimates for the pairs $(s_0,t_0)$ and $(s_0,-t_0)$, together with duality and complex interpolation of the Hilbert scale $H_h^r(\mathbb{R}^n)$, imply the same rapid estimate for the desired real orders. Since the residual hypothesis holds with every power of $h$, the interpolation constants and finite integer-order losses do not change the final $h^N$ conclusion. [/step] [step:Recover local kernel estimates from localized smoothing bounds] Assume the localized Sobolev mapping estimates. Let $E\subset\mathbb{R}^n\times\mathbb{R}^n$ be compact, and choose $\chi,\psi\in C_c^\infty(\mathbb{R}^n)$ such that $\chi(x)=1$ and $\psi(y)=1$ for every $(x,y)\in E$. Let $K_h^{\chi,\psi}:\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{C}$ be the smooth kernel of $\chi R_h\psi$, so that $K_h^{\chi,\psi}(x,y)=\chi(x)K_h(x,y)\psi(y)$. Fix multi-indices $\alpha,\beta\in\mathbb{N}_0^n$. Choose an integer $q>n/2+|\alpha|+|\beta|+1$. For each $y\in\operatorname{supp}\psi$, let $\delta_y$ denote the point mass at $y$, viewed as an element of $H_h^{-q}(\mathbb{R}^n)$. Semiclassical Sobolev estimates for point evaluation give constants $A>0$ and $C>0$, depending on $q,\beta,\psi,h_0$, such that \begin{align*} \|\partial_y^\beta(\psi\delta_y)\|_{H_h^{-q}}\leq Ch^{-A} \end{align*} uniformly for $y\in\operatorname{supp}\psi$. Applying the localized smoothing estimate with $s=q$, $t=q$, and exponent $N+A+1$ gives \begin{align*} \|\chi R_h\partial_y^\beta(\psi\delta_y)\|_{H_h^q}\leq C_Nh^{N+1} \end{align*} after changing the constant. The distribution $\chi R_h\partial_y^\beta(\psi\delta_y)$ is the function $x\mapsto(-1)^{|\beta|}\partial_y^\beta K_h^{\chi,\psi}(x,y)$. Since $q>n/2+|\alpha|$, the semiclassical Sobolev embedding estimate gives another fixed power loss $h^{-B}$ and therefore \begin{align*} \sup_{x\in\operatorname{supp}\chi}|\partial_x^\alpha\partial_y^\beta K_h^{\chi,\psi}(x,y)|\leq C_{E,\alpha,\beta,N}h^N \end{align*} for every $y\in\operatorname{supp}\psi$, after increasing the exponent used in the mapping estimate. On $E$ the cutoffs are identically one, so $K_h^{\chi,\psi}=K_h$ there. This proves the required local kernel estimate. [/step]