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The Liouville theorem for a class of Fourier multipliers and its connection to coupling

David Berger, René L. Schilling, Eugene Shargorodsky

Abstract

The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where $f$ is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes.

The Liouville theorem for a class of Fourier multipliers and its connection to coupling

Abstract

The classical Liouville property says that all bounded harmonic functions in , i.e.\ all bounded functions satisfying , are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator , such that the solutions to are Lebesgue a.e.\ constant (if is bounded) or coincide Lebesgue a.e.\ with a polynomial (if grows like a polynomial). The class of Fourier multipliers includes the (in general non-local) generators of Lévy processes. For generators of Lévy processes we obtain necessary and sufficient conditions for a strong Liouville theorem where is positive and grows at most exponentially fast. As an application of our results above we prove a coupling result for space-time Lévy processes.
Paper Structure (5 sections, 14 theorems, 71 equations)

This paper contains 5 sections, 14 theorems, 71 equations.

Table of Contents

  1. The proof of the Liouville theorem from BS21 revisited
  2. The Liouville theorem for polynomially bounded functions
  3. The Liouville theorem for slowly growing functions
  4. The Liouville theorem for rapidly growing functions
  5. Coupling

Key Result

Theorem 1

Let $\psi$ be the characteristic exponent of a Lévy process and denote by $\psi(D)$ the corresponding Fourier multiplier operator. Suppose $f \in L^\infty(\mathds{R}^n)$ is such that $\psi(D)f=0$ as a distribution, i.e. If $\left\{\eta \in \mathds{R}^n \mid \psi(\eta) = 0\right\} = \{0\}$, then $f \equiv \textrm{const}$ Lebesgue almost everywhere. Conversely, if $f \equiv \textrm{const}$ Lebesgu

Theorems & Definitions (33)

  • Theorem 1: Liouville; Ali-et-al20, BS21
  • Theorem 2
  • proof : Proof of Liouville's Theorem \ref{['th:1.1']}
  • Remark 3
  • Theorem 4
  • Remark 5
  • Example 6
  • Theorem 7
  • proof
  • Theorem 8: Liouville property for polynomially bounded functions
  • ...and 23 more