We prove sharp Carleman estimates and the corresponding unique continuation results for second-order real principal-type differential equations with critical potential (where is the dimension) across a noncharacteristic hypersurface under a pseudoconvexity assumption. Similarly, we prove unique continuation results for differential equations with potential in the Calderón uniqueness theorem's context under a curvature condition.
We also investigate ()-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.
David Dos Santos Ferreira. "Sharp Carleman estimates and unique continuation." Duke Math. J. 129 (3) 503 - 550, 15 September 2005. https://doi.org/10.1215/S0012-7094-05-12933-7