Unique continuation property for the anomalous diffusion and its application

The unique continuation for anomalous diffusion operators with fractional time derivative of order 1/2 is proved by deriving Carleman estimates for the operators. By applying operators which have a negative sign in front of the diffusion terms of anomalous diffusion operators, the anomalous diffusion operators are transformed to parabolic operators of order 4 in the space variables. The Carleman estimates are derived for these transformed operators considered as semi-elliptic operators. The usual Calderón uniqueness argument for semi-elliptic operators does not work due to the singularities in factors of the factorization of their principal parts. In order to have smooth factorizations of the operators, we took into account the lower order terms of the operators.