Sharp Lp Carleman estimates and unique continuation

Abstract

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 $V\in L_{\mathrm{loc}}^{n/2}$ (where $n\ge 3$ 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 ($L^p-L^{p\text{'}}$)-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.