Is it true that for $f,g \in H^s$, we have $$ \lVert fg \rVert_{H^s} \leq \lVert f \rVert_{H^s} \lVert g \rVert_{H^s}?$$ I think this is what the author is using on page 651 in this paper in the step $$ \lVert (\sigma^2 - \rho^2) \rVert_s \leq \lVert \sigma + \rho \rVert_s \lVert \sigma - \rho \rVert_s,$$ but I have never seen this type of inequality.
2026-04-03 17:55:23.1775238923
Bounding the $H^s$ norm of the product of two functions
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