Weakly well posed hyperbolic initial-boundary value problems with non characteristic boundary

We study the mixed initial-boundary value problem for a linear hyperbolic system with non characteristic boundary. We assume the problem to be “weakly” well posed, in the sense that a unique L2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of regularity. This is the case of problems that do not satisfy the uniform Kreiss- Lopatinski˘ı condition. Under the assumption of the loss of one tangential derivative, we obtain the Sobolev regularity of solutions, provided the data are sufficiently smooth