I am preparing talk on Beale-Kato-Majda blow condition for Euler/NS equation. I came across the following exercise (Exercise 2) in Tao's blog: https://terrytao.wordpress.com/2018/10/09/254a-notes-3-local-well-posedness-for-the-euler-equations/.
Let $a,b\geq 0$ be integers. For any $f \in H^{\infty}(\mathbb{R}^d \to \mathbb{R})$, show that $$|||\nabla^a f||\nabla^b g|||_{L^2}\leq_{a,b,d} ||f||_{L^{\infty}}||g||_{H^{a+b}}+||f||_{H^{a+b}}||g||_{L^{\infty}}$$ Hint: for $a=0$ or $b=0$ use Hölder, otherwise a suitable Littlewood-Paley decomposition.
For $a=b=0$ I just apply Hölder twice, for $a=0, b\neq 0$ I don't see how to move derivatives onto $f$.
There is a similar estimate in Prop 35 of https://terrytao.wordpress.com/2018/09/16/254a-notes-1-local-well-posedness-of-the-navier-stokes-equations/: $$||uv||_{H^s}<||u||_{H^s}||v||_{L^{\infty}}+||u||_{L^{\infty}}||v||_{H^s}||$$ and Tao uses the alternative norm $$||u||_{H^s}\sim_{d,s} ||P_{\leq 1}f||_{L^2}+\left(\sum_{N>1} N^{2s}||P_Nf||_{L^2}^2\right)^{1/2}.$$ But here we don't have an $H^s$ norm so I am not sure how to go about it. I could decompose $f=P_{\leq 1}f+\sum_{N>1}P_Nf$ and similarly for $g$, use the triangle inequality and multiply out, but I still have the derivatives. By Bernstein inequality, they disappear up to a power of N…