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

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