Calderón–Zygmund theory for strongly coupled linear system of nonlocal equations with Hölder-regular coefficient

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Abstract We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations ℒ A s ⁢ u = f {\mathcal{L}^{s}_{A}u=f} , where the operator ℒ A s {\mathcal{L}^{s}_{A}} is formally given by ℒ A s ⁢ u = ∫ ℝ n A ⁢ ( x , y ) | x - y | n + 2 ⁢ s ⁢ ( x - y ) ⊗ ( x - y ) | x - y | 2 ⁢ ( u ⁢ ( x ) - u ⁢ ( y ) ) ⁢ 𝑑 y . \mathcal{L}^{s}_{A}u=\int_{\mathbb{R}^{n}}\frac{A(x,y)}{|x-y|^{n+2s}}\frac{(x-% y)\otimes(x-y)}{|x-y|^{2}}(u(x)-u(y))\,dy. For 0 < s < 1 {0<s<1} and A : ℝ n × ℝ n → ℝ {A:\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{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 ) {A(\,\cdot\,,y)} is uniformly Holder continuous and inf x ∈ ℝ n ⁡ A ⁢ ( x , x ) > 0 {\inf_{x\in\mathbb{R}^{n}}A(x,x)>0} , then for f ∈ L loc p {f\in L^{p}_{\rm loc}} , for p ≥ 2 {p\geq 2} , the solution vector u ∈ H loc 2 ⁢ s - δ , p {u\in H^{2s-\delta,p}_{\rm loc}} for some δ ∈ ( 0 , s ) {\delta\in(0,s)} .