A Note on Unique Continuation for Parabolic Operators with Singular Potentials

We consider a class of heat-type differential operators. The coefficients in the lower-order terms are allowed to have critical singularities. These operators can be viewed as a perturbation of a simple model operator by a subcritical one. Under some conditions on the vector and scalar potentials in the critical part, we establish strong unique continuation theorems for such operators. For proof, we use a two-stage Carleman method. Firstly we derive a Carleman inequality for the model operators with critical potentials through an analysis of spectrum of some Schrodinger operators with compact resolvent. The obtained Carleman inequality at the first stage guarantees us to choose a weight function with higher singularity in a Carleman inequality at the second stage for the perturbed operators.