Localized smooth coordinate changes and cutoffs on Fourier $H^r$

If $U=\mathrm{\varnothing }$, then $V=\mathrm{\varnothing }$ because $\mathrm{\Phi }$ is a bijection. The chart operator and its inverse-chart analogue are then identically zero on every $H^r$, with operator norm zero. For the remaining chart argument assume $U\mathrm{\ne }\mathrm{\varnothing }$. By [F5], $\mathrm{supp}\chi$ is a compact subset of $U$. Choose fixed nonempty open patches $U_0⋐U$ and $V_0⋐V$ with $\mathrm{supp}\chi \subset U_0$ and $\mathrm{\Phi }(\overline{U_0})\subset V_0$. Put $\mathrm{\Psi }={\mathrm{\Phi }}^{-1}$. On these compact patches the derivatives of $\mathrm{\Phi }$ and $\mathrm{\Psi }$ through order two are bounded, and the absolute Jacobians are bounded above and away from zero. For $f\in L^2({\mathbb{R}}^d;\mathbb{C})$, the function $g(y)=\mathrm{\mid }\chi (\mathrm{\Psi }(y)){\mathrm{\mid }}^2\mathrm{\mid }f(y){\mathrm{\mid }}^2\mathrm{\mid }\det D\mathrm{\Psi }(y)\mathrm{\mid }$ on $\mathrm{\Phi }(U_0)$ belongs to $L^1$ by the compact Jacobian bound. Apply substitution to $g$ under $\mathrm{\Phi }{\mathrm{\mid }}_{U_0}$ and use $\mathrm{\mid }\det D\mathrm{\Psi }(\mathrm{\Phi }(x))\mathrm{\mid }\mathrm{\mid }\det D\mathrm{\Phi }(x)\mathrm{\mid }=1$ to obtain $\mathrm{\parallel }Kf{\mathrm{\parallel }}_2^2={\int }_{\mathrm{\Phi }(U_0)}\mathrm{\mid }\chi (\mathrm{\Psi }(y)){\mathrm{\mid }}^2\mathrm{\mid }f(y){\mathrm{\mid }}^2\mathrm{\mid }\det D\mathrm{\Psi }(y)\mathrm{\mid }dy\le C_0\mathrm{\parallel }\chi {\mathrm{\parallel }}_{\mathrm{\infty }}^2\mathrm{\parallel }f{\mathrm{\parallel }}_2^2$, where $C_0={\sup }_{\mathrm{\Phi }(\overline{U_0})}\mathrm{\mid }\det D\mathrm{\Psi }\mathrm{\mid }<\mathrm{\infty }$. If two input representatives differ on a null set $N$, their pullbacks differ only on $\mathrm{\Psi }(N\cap V)$, which is null by the null-set supplier applied to $\mathrm{\Psi }$. Thus the formula defines one complex-linear bounded operator on $L^2$ classes.

For $f\in H^2({\mathbb{R}}^d)$, the integer Fourier characterization identifies $f$ with its $W^{2,2}$ class with equivalent norms. Apply [F8] with ambient open set ${\mathbb{R}}^d$, restriction set $V_0$, $k=p=2$ and complex scalars: $\mathrm{\parallel }f{\mathrm{\mid }}_{V_0}{\mathrm{\parallel }}_{W^{2,2}(V_0)}\le \mathrm{\parallel }f{\mathrm{\parallel }}_{W^{2,2}({\mathbb{R}}^d)}$. The weak chain rule bounds $f\circ \mathrm{\Phi }$ in $W^{2,2}(U_0)$: its first derivatives use first input derivatives, and its second derivatives use first and second input derivatives with bounded chart coefficients. Leibniz bounds $\chi (f\circ \mathrm{\Phi })$, whose support is compact in $U_0$. Compact-support zero extension across only the ambient chart edges gives $\mathrm{\parallel }Kf{\mathrm{\parallel }}_{W^{2,2}({\mathbb{R}}^d)}\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{W^{2,2}({\mathbb{R}}^d)}$ for the same operator as in step 1.1.

The integer Fourier characterization turns the preceding bound into a bound on $H^2({\mathbb{R}}^d)$ for the same $L^2$ operator. On the nonzero frequency Hilbert space $L^2({\mathbb{R}}_{\xi }^d;\mathbb{C})$ let $M=M_{1+\mathrm{\mid }\xi {\mathrm{\mid }}^2}$. The multiplier is real and at least one, and Lebesgue measure is sigma-finite. Apply [F7] to $U={\mathcal{F}}_2^{-1}$ and $M$: the operator $B={\mathcal{F}}_2^{-1}M{\mathcal{F}}_2$ on $D(B)={\mathcal{F}}_2^{-1}D(M)$ is self-adjoint, with PVM $E_B(C)={\mathcal{F}}_2^{-1}{E}_M(C){\mathcal{F}}_2$. The scalar spectral measures are consequently transported by ${\mathcal{F}}_2$, so the multiplier calculus and square-integrability formula give $B\ge I$, $D(B^{r\mathrm{/}2})=\{f\in L^2:⟨\xi {⟩}^r{\mathcal{F}}_{2}f\in L^2\}$ and $\mathrm{\parallel }{B}^{r\mathrm{/}2}f{\mathrm{\parallel }}_2=\mathrm{\parallel }⟨\xi {⟩}^r{\mathcal{F}}_{2}f{\mathrm{\parallel }}_2$ for $0\le r\le 2$. The powers follow first for bounded truncated multipliers and then by their exact domain-norm limits. By [F9] and [F3], these domains are the bracket-weighted $H^r$ spaces with exactly their Fourier norms. The positive scale contract identifies its completed positive levels with these domains. Apply its common-operator interpolation to $K$ between levels zero and two to obtain the stated bounds for $0<r<2$; retain the endpoint bounds already proved. The inverse chart satisfies the same hypotheses on fixed compact inverse patches, so the identical argument applies there.

For a smooth multiplier with bounded derivatives through order two, the $L^2$ bound and global weak Leibniz bound give the same endpoint interpolation argument directly, without compact support. In all cases the extension to a positive-order space is the original $L^2$ formula restricted to that space, so local coordinate changes and cutoffs remain consistent under restriction and composition. This proves the claimed common-model localization.