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

In this paper, we establish Schauder’s estimates for the following non-local equations in \mR^d : $${\mathrm{\partial }}_{t}u={\mathcal{L}}_{\kappa ,\sigma }^{(\alpha )}u+b\cdot \mathrm{\nabla }u+f,u(0)=0,$$ where $\alpha \in (1\mathrm{/}2,2)$ 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 ${\mathcal{L}}_{\kappa ,\sigma }^{(\alpha )}$ 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_+ $isboundedfromaboveandbelow,$ \sigma:\mathbb R_+\times\mathbb R^{d}\to \mathbb R^d\otimes \mathbb R^d$isa$ \gamma $-order H"older continuous function in $x$ uniformly in $t$, and ${\nu }^{(\alpha )}$ is a singular non-degenerate $\alpha$-stable L'evy measure.