Regularity results for stable-like operators

For α∈[1,2)α∈[1,2) we consider operators of the formLf(x)=∫Rd[f(x+h)−f(x)−1(|h|⩽1)∇f(x)⋅h]A(x,h)|h|d+αdh and for α∈(0,1)α∈(0,1) we consider the same operator but where the ∇f   term is omitted. We prove, under appropriate conditions on A(x,h)A(x,h), that any solution u   to Lu=fLu=f will be in Cα+βCα+β if f∈Cβf∈Cβ.