$M,N \in \mathbb{D}$ or $2^{{\mathbb Z}}$ where $$\begin{aligned}
2^{\mathbb Z}& = {\left\{...,2^{-j},...,\frac{1}{4},\frac{1}{2},1,2,...,2^j,...\right\}}, \\ 
\mathbb{D} & = {\left\{\frac{1}{2},1,2,...,2^j,...\right\}} = 2^\mathbb{D}  \cup {\frac12} . \end{aligned}$$ This will be useful when defining Littlewood-Paley projections and paraproduct decompositions, see §\[Apx:LPProduct\]. Given a function $m \in L^\infty$, we define the Fourier multiplier $m({\nabla}) f$ by $$\begin{aligned}
(\widehat{m({\nabla})f})_k(\eta) =  m( (ik,i\eta) ) \hat{f}_k(\eta). \end{aligned}$$

We use the notation $f \lesssim g$ when there exists a constant $C > 0$ independent of the parameters of interest such that $f \leq Cg$ (we analogously define $f \gtrsim g$). Similarly, we use the notation $f \approx g$ when there exists $C > 0$ such that $C^{-1}g \leq f \leq Cg$. We sometimes use the notation $f \lesssim_{\alpha} g$ if we want to emphasize that the implicit constant depends on some parameter $\alpha$. We will denote the $l^1$ vector norm ${\left\vertk,\eta\right\vert} = {\left\vertk\right\vert} + {\left\vert\eta\right\vert}$, which by convention is the norm taken in our work. Similarly, given a scalar or vector in ${\mathbb R}^n$ we denote $$\begin{aligned}
{\left\langle v \right\rangle} = \left( 1 + {\left\vertv\right\vert}^2 \right)^{1/2}. \end{aligned}$$ We use a similar notation to denote the $x$ (or $z$) average of a function: $<f> = \frac{1}{2\pi}\int f(x,y) dx = f_0$. We 