On page 339 of Taos's Nonlinear Dispersive Equations, he makes the following estimate (in the context of proving a particular fractional Leibniz rule).
Let $M,N$ be dyadic numbers and $P_N$ be the Littlewood-Paley multiplier given by $\widehat{P_N}f(\xi) := (\varphi(\xi/N) - \varphi(2\xi/N))\widehat{f}(\xi)$ where $\varphi(\xi)$ is a smooth bump function adapted to the ball $\{\xi \in \mathbb{R}^d : |\xi|\leq 2\}$ and equals $1$ on the ball $\{\xi \in \mathbb{R}^d : |\xi|\leq 2\}$.
Then
\begin{align}\sum_{M>N/8} ||P_N((P_Mf)g)||_{L^2_x} &\leq C \sum_{M>N/8} ||(P_Mf)g||_{L^2_x} \\ &\leq C ||g||_{L^\infty_x} \sum_{M \geq C_0N} M^{-s}||P_M f||_{H^s_x} \end{align} (The $H^s$ in the second line is not in the book but was a correction in the errata) I think I understand the first inequality, the operator $P_N$ is bounded so we can always write $||P_N(f)||_{L^2_x} \leq ||f||_{L^2_x}$. The second inequality is what is causing me trouble. I'm pretty sure he applied Holder's inequality to $P_Mf$ and $g$ to get $$||(P_Mf)g||_{L^2_x} \leq ||g||_{L^\infty_x} ||P_Mf||_{L^2_x}$$ However I'm not sure where the $H^s$ and $M^{-s}$ factor comes from. I imagine some type of bernstein inequality might be involved but in Tao's book we only have bernstein inequalities involving the homogeneous multiplier $|\nabla|^s$ so it seems awkward to try to apply them when nonhomogenous sobolev spaces are involved.
Also $C$ and $C_0$ are constants.
Thanks.
It looks like not Bernstein but rather the Jackson (aka direct) estimates are involved. Note that for terms like $P_Mf$ there is no difference between homogeneous and inhomogeneous Sobolev norms.