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The $$X\theta^ s$$ spaces and unique continuation for solutions to the semilinear wave equation. (English) Zbl 0853.35017
Summary: The aim of this paper is twofold. First, we initiate a detailed study of the so-called $$X^s_\theta$$ spaces attached to a partial differential operator. This includes 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 $$L^p\to L^q$$ Carleman estimates, derived using the $$X^s_\theta$$ 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.

MSC:
 35B60 Continuation and prolongation of solutions to PDEs 35L70 Second-order nonlinear hyperbolic equations
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References:
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