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We will present a unique continuation result for solutions of second order differential equations of real principal type $P(x,D)u+V\left(x\right)u=0$ with critical potential $V$ in $L^{n/2}$ (where $n$ is the number of variables) across non-characteristic pseudo-convex hypersurfaces. To obtain unique continuation we prove $L^p$ Carleman estimates, this is achieved by constructing a parametrix for the operator conjugated by the Carleman exponential weight and investigating its $L^p-L^{p^{\text{'}}}$ boundedness properties.

This note reports on recent results on the anisotropic Calderón problem obtained in a joint work with Carlos E. Kenig, Mikko Salo and Gunther Uhlmann []. The approach is based on the construction of complex geometrical optics solutions to the Schrödinger equation involving phases introduced in the work [] of Kenig, Sjöstrand and Uhlmann in the isotropic setting. We characterize those manifolds where the construction is possible, and give applications to uniqueness for the corresponding anisotropic...