Schauder Estimates for Nonlocal Equations with Singular Lévy Measures

Abstract

In this paper, we establish Schauder’s estimates for the following non-local equations in $\mathbb {R}^{d}$ :

$$ \partial _t u=\mathscr {L}^{(\alpha )}_{\kappa ,\sigma } u+b\cdot \nabla u+f,\ u(0)=0, $$

where $\alpha \in (1/2,2)$ and $b:\mathbb {R}_+\times \mathbb {R}^d\rightarrow \mathbb R$ is an unbounded local $\beta$-order Hölder function in x uniformly in t, and $\mathscr {L}^{(\alpha )}_{\kappa ,\sigma }$ is a non-local $\alpha$-stable-like operator with form:

$$ \mathscr {L}^{(\alpha )}_{\kappa ,\sigma }u(t,x):=\int _\mathbb {R}^d\Big (u(t,x+\sigma (t,x)z)-u(t,x)-\sigma (t,x)z^{(\alpha )}\cdot \nabla u(t,x)\Big )\kappa (t,x,z)\nu ^{(\alpha )}(\mathord {\textrm{d}} z), $$

where $z^{(\alpha )}=z\textbf{1}_{\alpha \in (1,2)}+z\textbf{1}_{|z|\le 1}\textbf{1}_{\alpha =1}$, $\kappa :\mathbb {R}_+\times \mathbb {R}^{2d}\rightarrow \mathbb {R}_+$ is bounded from above and below, $\sigma :\mathbb {R}_+\times \mathbb {R}^{d}\rightarrow \mathbb {R}^d\otimes \mathbb {R}^d$ is a $\gamma$-order Hölder continuous function in x uniformly in t, and $\nu ^{\alpha )}$ is a singular non-degenerate $\alpha$-stable Lévy measure.