Back to Search
Start Over
The Liouville theorem for a class of Fourier multipliers and its connection to coupling.
- Source :
-
Bulletin of the London Mathematical Society . Jul2024, Vol. 56 Issue 7, p2374-2394. 21p. - Publication Year :
- 2024
-
Abstract
- The classical Liouville property says that all bounded harmonic functions in Rn$\mathbb {R}^n$, that is, all bounded functions satisfying Δf=0$\Delta f = 0$, are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator m(D)$m(D)$, such that the solutions f$f$ to m(D)f=0$m(D)f=0$ are Lebesgue a.e. constant (if f$f$ is bounded) or coincide Lebesgue a.e. with a polynomial (if f$f$ is polynomially bounded). 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$f$ 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. [ABSTRACT FROM AUTHOR]
Details
- Language :
- English
- ISSN :
- 00246093
- Volume :
- 56
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Bulletin of the London Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 178296460
- Full Text :
- https://doi.org/10.1112/blms.13060