Schauder's estimates for nonlocal equations with singular Lévy measures

Abstract

In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $${\partial }_{t}u={\mathcal{L}}_{\kappa ,\sigma }^{(\alpha )}u+b\cdot \nabla u+f,u(0)=0,$$ where $\alpha \in (1/2,2)$ and $b:{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to \mathbb{R}$ is an unbounded local $\beta$-order Hölder 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}^{(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 $z^{(\alpha )}=z{1}_{\alpha \in (1,2)}+z{1}_{|z|\le 1}{1}_{\alpha =1}$, $\kappa :{\mathbb{R}}_{+}\times {\mathbb{R}}^{2d}\to {\mathbb{R}}_{+}$ 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 Hölder continuous function in $x$ uniformly in $t$, and ${\nu }^{(\alpha )}$ is a singular non-degenerate $\alpha$-stable Lévy measure.

References in corpus (3)

• Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts
• Schauder's estimate for nonlocal kinetic equations and its applications
• Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $\alpha$-stable processes