method. However, in our case the I-method cannot be directly applied. It requires a local theory in $L^2$-based Sobolev spaces whose norms can be controlled by the modified energy. However, there is no such local theory for wave data below $L^2$ for the Zakharov system. We therefore must subtract out the linear flow in order to solve in Sobolev spaces. This leaves a difference equation, which is no longer expected to satisfy an almost-conservation law.

On the other hand, if we attempt to apply the high-low method alone, we are required to obtain estimates of the form $$\| u W(t)(n_0^\pm) \|_{X^{1, -\frac12, 1}} \lesssim \|n_0^\pm\|_{Y^{r,\frac12,1}} \| u \|_{X^{1, \frac12, 1}}.$$ Here $W(t)(n_0^\pm)$ is the linear wave flow. The norms which appear here are defined below. The relevant point for the moment is that the left-hand side is a norm with Sobolev regularity one, while the wave flow is rougher than $L^2$. For such rough wave data, it seems impossible to obtain the necessary estimate. Randomization of the wave data does not appear to improve matters. In the $H^s \times H^r$ local theory, this case corresponds to the (unattainable) $s > r +1$ regime.

In our work, we begin by splitting 