\| {\operatorname{I}\!}u_0 \|_{H^1} + \| n_0^{\pm \text{L}} \|_{L^2} \right) ^{-2-}.$$

Here the smoothing operator ${\operatorname{I}\!}: H^s \to H^1$ is defined by $${\widehat}{{\operatorname{I}\!}f}(\xi) = m(\xi) {\widehat}{f}(\xi), \quad \text{where} \quad m(\xi) = \begin{cases}
     1 &\text{for  } |\xi| <  {N_{\text{I}}}, \\
     ({N_{\text{I}}}/|\xi|)^{1-s} &\text{for  } |\xi| > 2{N_{\text{I}}}.
    \end{cases}$$ The multiplier $m$ is taken to be smooth and non-increasing in $|\xi|$. The multiplier, and hence ${\operatorname{I}\!}\;$ itself, is dependant on the choice of ${N_{\text{I}}}$, but we elect to suppress this in the notation for simplicity.

For initial data satisfying $\| u_0\|_{H^s} = K$ and $\sup_k | {\widehat}{n_0^{\pm \text{L}}}(k)| \leq K {\langle}k {\rangle}^{-\beta}$, we may choose [^1] $$\delta \approx  \Bigl[  K {N_{\text{I}}}^{1-s} +  K {N_{\text{HL}}}^{\frac12 -\beta} \Bigr]^{-2-} =: N^{-2-}.$$ and obtain the bound $$\label{eq:normBound} \delta^{\frac12-} \left( \| {\operatorname{I}\!}u^1 \|_{X^{1,\frac12,1}_\delta} + \| n^{\pm1} \|_{Y^{0,\frac12,1}_{\pm,\delta}} \right) \lesssim 1.$$ To continue the argument, we need a local theory for the difference equation which results when we subtract this solution $(u^1,n^{\pm1})$ from the solution for the Zakharov system with the full initial data $(u_0,n_0)$. This local theory is given by Proposition \[localtheory\] below. Essentially, it says that as long as $|{\widehat}{n_0^{\pm \text{H}}}(k)| \leq C_1 {\langle}k {\rangle}^{-\beta}$, the difference equation can be solved on $[0,\delta]$, and the $H^s \times L^2$ 