  We establish new bounds of the Sobolev norms of solutions of semilinear wave
equations for data lying in the Hs, s<1, closure of compactly supported data
inside a ball of radius R, with R a fixed and positive number. In order to do
that we perform an analysis in the neighborhood of the cone, using an almost
Shatah-Struwe estimate, an almost conservation law and some estimates for
localized functions: this allows to prove a decay estimate and establish a low
frequency estimate of the position of the solution. Then, in order to establish
a high frequency estimate of the position and an estimate of the velocity, we
use this decay estimate and another almost conservation law.
