  By proving a weighted contraction estimate in uniformly local Sobolev spaces
for the flow of gravity water waves, we show that this nonlocal system is in
fact pseudo-local in the following sense: locally in time, the dynamic far away
from a given bounded region has a small effect on that region (again, in a
sense that we will make precise in the article). Our estimate on the flow also
implies a new spatial decay property of the waves. To prove this result, we
establish a paradifferential calculus theory in uniformly local Sobolev spaces
with weights.
