Calderon-Zygmund theory for strongly coupled linear system of nonlocal equations with Holder-regular coefficient

We extend the Calderón-Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ℒ^s_A u = f, where the operator ℒ^s_A is formally given by ℒ^s_Au = ∫_ℝ^nA(x, y)/| x-y| ^n+2s(x-y)⊗ (x-y)/| x-y| ^2(u(x)-u(y))dy. For 0 < s < 1 and A:ℝ^n×ℝ^n→ℝ taken to be symmetric and serving as a variable coefficient for the operator, the system under consideration is the fractional version of the classical Navier-Lamé linearized elasticity system. The study of the coupled system of nonlocal equations is motivated by its appearance in nonlocal mechanics, primarily in peridynamics. Our regularity result states that if A(·, y) is uniformly Holder continuous and inf_x∈ℝ^nA(x, x) > 0, then for f∈ L^p_loc, for p≥ 2, the solution vector u∈ H^2s-δ,p_loc for some δ∈ (0, s).