Non-local operators with low singularity kernels: regularity estimates and martingale problem

We consider the linear non-local operator $\mathcal{L}$ denoted by $$\mathcal{L}u(x)={\int }_{{\mathbb{R}}^d}(u(x+z)-u(x))a(x,z)J(z)dz\mathrm{.}$$ Here $a(x,z)$ is bounded and $J(z)$ is the jumping 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 $\mathcal{L}$, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with $\mathcal{L}$. By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.