Schauder's estimates for nonlocal equations with singular Lévy measures

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

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