Regularity of weakly well posed hyperbolic mixed problems with characteristic boundary

We study the mixed initial-boundary value problem for a linear hyperbolic system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a unique L^2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss–Lopatinskiı̆ condition in the hyperbolic region of the frequency domain. Under the assumption of the loss of one conormal derivative we obtain the regularity of solutions, in the natural framework of weighted anisotropic Sobolev spaces, provided the data are sufficiently smooth.