Regular solution to hyperbolic boundary value problems without regularization

In this article we study the regularity of the solution to hyperbolic boundary value problems defined in the half-space. Using regularization methods, mollification or non characteristic regularization, the regularity of such solutions is a well-known fact in the literature. However, we propose here two direct approaches to show this regularity without these regularization steps. The first method is based upon Hille-Yosida theorem, while the second one is based upon the duality method of Lax-Phillips. An interesting point of these procedures is that both rely on the injectivity of some dual operator. An other point of interest is that both methods are robust enough to handle, without any substantial modification, with space variable coefficients and to characteristic problems, while the existing methods in the literature sometimes need some rather deep modifications.