In this talk, I will first present recent results on global Lp Carleman estimates for the Laplacian, as well as their application to the quantification of unique continuation for solutions of the equation Δu=Vu+W1​⋅∇u+div(W2​u), in terms of the norms of the potentials V∈Lq0​, W1​∈Lq1​, and W2​∈Lq2​. This is a joint work with my PhD advisors, Belhassen Dehman and Sylvain Ervedoza.

In the second part of the talk, I will present results on the quantification of unique continuation for solutions of the same equation Δu=Vu+W1​⋅∇u+div(W2​u). Using Thomas Wolff's lemma on Euclidean measures and a refined version of Carleman estimates, we obtain quantification results for the unique continuation of solutions u in terms of the norms of the potentials, including first-order potentials that are more singular in the limit integrability class. In particular, we consider the case where W1​∈Lq1​ and W2​∈Lq2​, with q1​>d and q2​>d.

This is a joint work with Pedro Caro and Sylvain Ervedoza.