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arXiv · 2604.02357

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

David Berger·Rene L. Schilling
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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.

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David Berger, Rene L. Schilling. 2026-03-16. On the Unique Continuation Principle for a Class of Translation Invariant Nonlocal Operators. https://arxiv.org/abs/2604.02357

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