in $s$ (and obtain enough gain) for all low frequency inputs.

Without (\[assm2\]), one has to study low frequency inputs on a case by case basis; moreover, the point $(0,0)$, which can now be resonant, can actually be very degenerate, meaning it is possible to have$(\nabla_{\eta}^{\alpha}\Phi)(0,0)=0$ for $|\alpha|\leq 3$ (which is clearly a disadvantage in terms of stationary phase analysis). One example is when$$\Phi(\xi,\eta)=\sqrt{|\xi|^{2}+1}-\sqrt{2|\xi-\eta|^{2}+4}+\sqrt{|\eta|^{2}+1}=\xi\cdot\eta/2+O(|\xi|^{4}+|\eta|^{4}).$$ To see the effect of this degeneracy, we consider a “rescaled Schwartz” component $f_{j,-j}$ localized in frequency at scale $\sim 2^{-j}$ and in space at scale $\sim2^j$. Suppose that $|s|\sim 2^{m}$ in (\[duhamel\]), and that the inputs are $$\widehat{f_{\mu}}(s,\xi-\eta)=2^{cj}\chi(2^{j}(\xi-\eta)),\qquad \widehat{f_{\nu}}(s,\eta)=2^{cj}\chi(2^{j}\eta)$$ with $j=m/4$ and some constant $c$, then in the region where $|\eta|\sim|\xi-\eta|\sim 2^{-j}$ and $|\xi|\sim 2^{-3j}$, we have $|\Phi|\lesssim 2^{-m}$, so the oscillation factor $e^{is\Phi}$ is irrelevant, thus the output will be a rescaled Schwartz function supported at frequency scale $|\xi|\sim 2^{-3j}$ with $L_{\xi}^{\infty}$ norm bounded by$$2^{m}2^{-3j}2^{2cj}=2^{(3j)(2c+1)/3}.$$ If we start with $c=0$ (imagine cutting off a Schwartz function at scale $|\xi|\sim 2^{-j}$), then in finitely many iterations we obtain a profile with form $2^{cj}\chi(2^{j}\xi)$, where $c$ can be arbitrarily close to $1$ (which is the unique fixed point of the map $c\mapsto (2c+1)/3$). Therefore, if we 