Non-local operators with low singularity kernels regularity estimates and martingale problem (Guohuan Zhao)

We consider the linear non-local operator L denoted by Lu(x) = ∫Rd (u(x + z) − u(x)) a(x, z)J(z) dz. Here a(x, z) is bounded and J(z) is the jump kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scal­ing. The aim of this work is twofold. Firstly, we introduce generalized Orlicz–Besov spaces tailored to accommodate the analysis of elliptic equations associated with L, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with L. By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Finally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz–Besov spaces.

Publication:

Mathematische Annalen (2025) 393:1881–1937

https://doi.org/10.1007/s00208-025-03199-2

Author:

Guohuan Zhao

SKLMS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

gzhao@amss.ac.cn

Eryan Hu

Center for Applied Mathematics and KL-AAGDM, Tianjin University, Tianjin 300072, China

eryan.hu@tju.edu.cn