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

TL;DR

This paper establishes necessary and sufficient conditions for the unique continuation property of Lévy operators, linking it to their resolvent, and provides new proofs for fractional Laplace and Bernstein functions of the discrete Laplacian.

Contribution

It introduces a comprehensive criterion for UCP in Lévy operators and connects it to the resolvent, with applications to fractional Laplace and Bernstein functions.

Findings

Established criteria for UCP for Lévy operators.

Connected UCP to the resolvent of the operators.

Provided new elementary proofs for fractional Laplace and Bernstein functions.

Abstract

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 L\'evy operators. We prove a connection between the UCP of the L\'evy 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.