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

David Berger, Rene L. Schilling

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 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.

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

Abstract

The unique continuation property (UCP) for an operator says that, if holds on an open set , then one has 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.

Paper Structure

This paper contains 4 sections, 11 theorems, 52 equations, 1 figure.

Table of Contents

  1. A class of nonlocal operators
  2. The unique continuation principle for Lévy operators
  3. The unique continuation principle for resolvents and semigroups
  4. The uniform continuation principle for discrete Lévy operators

Key Result

Lemma 1.2

If the underlying Banach space $\mathcal{X}$ is $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: The intersection of $B_\delta(x_0)$ with the shifted ball $x'+B_\delta(x_0) = B_\delta(x_0+x')$, $|x'|<\frac{1}{2}\delta$, contains the balls $B_\eta(x_0)$ and $B_\eta(x_0+x')$ for any $\eta\in \left(0,\frac{1}{2}\delta\right)$.

Theorems & Definitions (28)

  • Definition 1.1
  • Lemma 1.2
  • proof
  • Definition 2.1
  • Lemma 2.2
  • Theorem 2.3
  • proof
  • Lemma 2.4
  • proof
  • Example 2.5
  • ...and 18 more