Non-symmetric stable operators: Regularity theory and integration by parts

We study solutions to Lu=f in Ω⊂Rⁿ, being L the generator of any, possibly non-symmetric, stable Lévy process. On the one hand, we study the regularity of solutions to Lu=f in Ω, u=0 in Ω^c, in C^1,α domains Ω. We show that solutions u satisfy u/d^γ∈C^ε_∘(Ω‾), where d is the distance to ∂Ω, and γ=γ(L,ν) is an explicit exponent that depends on the Fourier symbol of operator L and on the unit normal ν to the boundary ∂Ω. On the other hand, we establish new integration by parts identities in half spaces for such operators. These new identities extend previous ones for the fractional Laplacian, but the non-symmetric setting presents some new interesting features. Finally, we generalize the integration by parts identities in half spaces to the case of bounded C^1,α domains. We do it via a new efficient approximation argument, which exploits the Hölder regularity of u/d^γ. This new approximation argument is interesting, we believe, even in the case of the fractional Laplacian.