= High-frequency reciprocal parametrix kernel
{title2=$b(\xi)=(1-\psi(\xi))/P(\xi)$}
For a <polynomial> with no sufficiently large real zero, a compact low-frequency <cutoff function> removes its remaining zeros. A reciprocal satisfying $|\partial^\alpha b|\lesssim\langle\xi\rangle^{-m-\delta|\alpha|}$, $m\geq0$, $0<\delta\leq1$, defines a <Fourier multiplier> mapping $H^s$ to $H^{s+m}$. Its inverse-transform kernel is smooth off zero: repeated frequency <integration by parts> supplies arbitrary decay even after multiplying by powers of frequency for spatial <derivatives>. Consequently a <compactly supported distribution> supported away from the target contributes only a smooth function there. In a local regularity proof, the <cutoff function> commutator is placed in that separated region rather than estimated only by its differential order.
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