TY - JOUR

T1 - The Liouville theorem for a class of Fourier multipliers and its connection to coupling

AU - Berger, David

AU - Schilling, René L.

AU - Shargorodsky, Eugene

N1 - Publisher Copyright:
© 2024 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.

PY - 2024/5/9

Y1 - 2024/5/9

N2 - The classical Liouville property says that all bounded harmonic functions in (Formula presented.), that is, all bounded functions satisfying (Formula presented.), are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator (Formula presented.), such that the solutions (Formula presented.) to (Formula presented.) are Lebesgue a.e. constant (if (Formula presented.) is bounded) or coincide Lebesgue a.e. with a polynomial (if (Formula presented.) 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 (Formula presented.) 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.

AB - The classical Liouville property says that all bounded harmonic functions in (Formula presented.), that is, all bounded functions satisfying (Formula presented.), are constant. In this paper, we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator (Formula presented.), such that the solutions (Formula presented.) to (Formula presented.) are Lebesgue a.e. constant (if (Formula presented.) is bounded) or coincide Lebesgue a.e. with a polynomial (if (Formula presented.) 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 (Formula presented.) 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.

UR - https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/blms.13060

UR - http://www.scopus.com/inward/record.url?scp=85192534826&partnerID=8YFLogxK

U2 - 10.1112/blms.13060

DO - 10.1112/blms.13060

M3 - Article

SN - 0024-6093

JO - BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

JF - BULLETIN OF THE LONDON MATHEMATICAL SOCIETY

ER -