TheXsθspaces and unique continuation for solutions to the semilinear wave equation

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Abstract
The aim of this paper is twofold. First, we initiate a detailed study of the so-called Xs θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and Lp (Lq ) 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 Lp → Lq Carleman estimates, derived using the Xs θ 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 characteristic set satisfies a curvature conditions.