Schauder's estimates for nonlocal equations with singular L\'evy measures

Abstract

In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(\alpha)}_{\kappa,\sigma} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $\alpha\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb R^d\to\mathbb R$ is an unbounded local $\beta$-order H\"older function in $ x $ uniformly in $ t $, and $\mathscr L^{(\alpha)}_{\kappa,\sigma}$ is a non-local $\alpha$-stable-like operator with form: \begin{align*} {\mathscr L}^{(\alpha)}_{\kappa,\sigma}u(t,x):=\int_{\mathbb R^d}\Big(u(t,x+\sigma(t,x)z)-u(t,x)-\sigma(t,x)z^{(\alpha)}\cdot\nabla u(t,x)\Big)\kappa(t,x,z)\nu^{(\alpha)}(\mathord{{\rm d}} z), \end{align*} where $z^{(\alpha)}=z\mathbf{1}_{\alpha\in(1,2)}+z\mathbf{1}_{|z|\leq 1}\mathbf{1}_{\alpha=1}$, $ \kappa:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R_+ $ is bounded from above and below, $ \sigma:\mathbb R_+\times\mathbb R^{d}\to \mathbb R^d\otimes \mathbb R^d$ is a $ \gamma $-order H\"older continuous function in $ x $ uniformly in $ t $, and $ \nu^{(\alpha)} $ is a singular non-degenerate $ \alpha $-stable L\'evy measure.