= Tame Sobolev product estimate
{title2=$\|fg\|_{H^k}\lesssim\|f\|_\infty\|g\|_{H^k}+\|g\|_\infty\|f\|_{H^k}$}
For an integer $k\geq0$ and sufficiently regular bounded functions, $\|fg\|_{H^k}\leq C_k(\|f\|_\infty\|g\|_{H^k}+\|g\|_\infty\|f\|_{H^k})$. The <Leibniz rule> and derivative interpolation give the estimate. In a differentiated <semilinear wave equation>, this separates a controlled low-order coefficient from an arbitrarily high <Sobolev norm>, allowing propagation of smoothness after a fixed base norm has been controlled.
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