= Derivative-ratio Sobolev gain for a polynomial operator
{title2=$Q(D)u\in H^s_{\mathrm{loc}}\Longrightarrow u\in H^{s+\delta N}_{\mathrm{loc}}$}
Suppose a degree-$N$ <polynomial> obeys $|\partial^\alpha Q(\xi)|\leq C_\alpha|\xi|^{-\delta|\alpha|}|Q(\xi)|$ at large real frequency, with $\delta>0$. A nonzero constant <derivative> of total order $N$ gives $|Q|\gtrsim\langle\xi\rangle^{\delta N}$. Reciprocal differentiation then bounds $\partial^\alpha(1/Q)$ by $\langle\xi\rangle^{-\delta N-\delta|\alpha|}$. The <high-frequency reciprocal parametrix kernel> therefore gains $\delta N$ Sobolev <derivatives>. Localize the <distribution> by a <cutoff function> equal to one near the target; the commutator lies outside that target and the reciprocal kernel makes its contribution smooth. This proves the displayed gain without demanding global regularity of the original <distribution>.
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