We prove sharp $L^p$ 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 $V \in L^{n/2}_{\rm loc}$ (where $n \geq 3$ is the dimension) across a noncharacteristic hypersurface under a pseudoconvexity assumption. Similarly, we prove unique continuation results for differential equations with potential in the Calderón uniqueness theorem's context under a curvature condition.
¶ We also investigate ( $L^p\,{-}L^{p'}$ )-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.