$L^1$-norm. Controlling this norm leads us to make most of our analysis in the physical space rather than in the Fourier space. We believe this gives a clearer picture and accounts better for the finite speed of propagation inherent in dispersive equations[^6].

In particular, once we introduce localization in physical space, the main problem becomes to bound uniformly the linear profile on a spatial dyadic ring in $xL^2$. Assume to simplify that the initial data is only concentrated in the ball of radius $1$ and that we are looking at a distance $R>>1$. The bound follows from three distinct ingredients:

1.  First, by variation on the finite speed of propagation principle, one can see that, for times $t\lesssim R$, the solution at $R$ is only strongly influenced by the solution at nearby locations (at distance smaller than $R/2$). Since it starts small and it interacts nonlinearly, it remains so.

2.  Second, by decay property of the solutions, all the interactions after a time $t>R^{1+\delta}$ add up in $L^2$ to a size smaller than $R^{1-\delta/100}$ and thus are acceptable.

3.  Finally, one only needs to consider a small portion of space-time when $R\sim x$, $R\le t\le R^{1+\delta}$. This is of course 