We consider the linear non-local operator \(\mathcal {L}\) denoted by \(\begin{aligned} \mathcal {L} u (x) = \int _{\mathbb {R}^d} \left( u(x+z)-u(x)\right) a(x,z)J(z)\,\textrm{d} z. \end{aligned}\) 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 \(\mathcal {L}\) , 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. 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.
