Abstract
We prove sharp Lp Carleman estimates and the corresponding unique continuation results for second-order real principal-type differential equations P(x,D)u+V(x)u=0 with critical potential V∈Llocn/2 (where n≥3 is the dimension) across a noncharacteristic hypersurface under a pseudoconvexity assumption. Similarly, we prove unique continuation results for differential equations with potential in the Calderon uniqueness theorem's context under a curvature condition. We also investigate (Lp-Lp')-estimates for non-self-adjoint pseudodifferential operators under a curvature condition on the characteristic set and develop the natural applications to local solvability for the corresponding operators with potential.
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