Regularity results for stable-like operators

Abstract

For $\alpha \in [1,2)$ we consider operators of the form $$Lf(x)={\int }_{R^d}[f(x+h)-f(x)-1_{(|h|\le 1)}\nabla f(x)\cdot h]\frac{A(x,h)}{|h{|}^{d+\alpha }}$$ and for $\alpha \in (0,1)$ we consider the same operator but where the $\nabla f$ term is omitted. We prove, under appropriate conditions on $A(x,h)$, that the solution $u$ to $Lu=f$ will be in $C^{\alpha +\beta }$ if $f\in C^{\beta }$.

References in corpus (1)

• Uniqueness for the martingale problem associated with pure jump processes of variable order