The X(0)(5) spaces and unique continuation for solutions to the semilinear wave equation

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X(theta)(s) spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L(p)(L(q)) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L(p) --> L(q) Carleman estimates, derived using the X(theta)(s) spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that the characteristic set satisfies a curvature condition.