On the Unique Continuation Principle for a Class of Translation Invariant Nonlocal Operators

The unique continuation property (UCP) for an operator $A$ says that, if $Au=0=u$ holds on an open set $G$, then one has $u=0$ everywhere. We establish necessary and sufficient conditions for the UCP for the class of Lévy operators. We prove a connection between the UCP of the Lévy operator and its resolvent. Our results are applied to obtain a new elementary proof of the UCP for the fractional Laplace operator, and for certain functions (Bernstein functions) of the discrete Laplace operator.