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

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