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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| Sujets: | |
| Accès en ligne: | https://arxiv.org/abs/2304.14056 |
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Table des matières:
- 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 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.