  We consider the linear non-local operator $\mathcal{L}$ denoted by \[
\mathcal{L} u (x) = \int_{\mathbb{R}^d} \left(u(x+z)-u(x)\right) a(x,z)J(z)\,d
z. \] Here $a(x,z)$ is bounded and $J(z)$ is the jumping kernel of a L\'evy
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.
