Regularity results for stable-like operators

For α ∈ [1, 2) we consider operators of the form Lf (x) = Rd f (x + h) −f (x)− 1(|h| 1)∇f (x) · h A(x, h) |h|d+α dh and for α ∈ (0, 1) we consider the same operator but where the ∇f term is omitted. We prove, under appropriate conditions on A(x, h), that any solution u to Lu = f will be in Cα+β if f ∈ Cβ. © 2009 Elsevier Inc. All rights reserved.