Calderon-Zygmund type estimates for nonlocal PDE with Hölder continuous kernel

Abstract

We study interior Lp-regularity theory, also known as Calderon-Zygmund theory, of the equation〈Lsu,φ〉:=∫Rn∫RnK(x,y)(u(x)−u(y))(φ(x)−φ(y))|x−y|n+2sdxdy=〈f,φ〉,∀φ∈Cc∞(Rn). We prove that for s∈(0,1), t∈[s,2s], p∈[2,∞), K an elliptic, symmetric, and K(⋅,y) is uniformly Hölder continuous, the solution u belongs to Hloc2s−t,p(Ω) as long as 2s−t<1 and f∈(H00t,p′(Ω))⁎.The increase in differentiability and integrability is independent of the Hölder coefficient of K. For example, in the event that f∈Llocp, we can deduce that the solution u∈Hloc2s−δ,p for any δ∈(0,s] as long as 2s−δ<1. This regularity result is different from its classical analogue for divergence-form equations div(K¯∇u)=f where a Cγ-Hölder continuous coefficient K¯ only allows solutions in H1+γ. In fact, the regularity estimates we prove are another manifestation of the differential stability effects of nonlocal equations of the above that are observed by many authors – only that in our case we do not get a “small” differentiability improvement, but all the way up to min⁡{2s−t,1}.The proof argues by comparison with the (much simpler) equation〈Ldiags,tu,φ〉:=∫RnK(z,z)(−Δ)t2u(z)(−Δ)2s−t2φ(z)dz=〈g,φ〉,∀φ∈Cc∞(Rn), and showing that as long as K is Hölder continuous and s,t,2s−t∈(0,1) then the “commutator” Lsu−Ldiags,tu behaves like a lower order operator.