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

We extend the Calderón–Zygmund theory for nonlocal equations to strongly coupled system of linear nonlocal equations <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mi>f</m:mi> </m:mrow> </m:math> {LˢAu=f} , where the operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> </m:math> {LˢA} is formally given by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mrow> <m:msubsup> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>A</m:mi> <m:mi>s</m:mi> </m:msubsup> <m:mo>⁢</m:mo> <m:mi>u</m:mi> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:msub> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:msub> <m:mrow> <m:mfrac> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> </m:mrow> </m:msup> </m:mfrac> <m:mo>⁢</m:mo> <m:mfrac> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>⊗</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>-</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mn>2</m:mn> </m:msup> </m:mfrac> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>-</m:mo> <m:mrow> <m:mi>u</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo rspace="4.2pt" stretchy="false">)</m:mo> </m:mrow> <m:mo>⁢</m:mo> <m:mrow> <m:mo>𝑑</m:mo> <m:mi>y</m:mi> </m:mrow> </m:mrow> </m:mrow> </m:mrow> <m:mo>.</m:mo> </m:mrow> </m:math> LˢAu=∫_{Rⁿ}{A(x,y)}{|x-y|ⁿ⁺²ˢ}{(x-% y)⊗(x-y)}{|x-y|²}(u(x)-u(y))\,dy. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>s</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<s<1} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {A:Rⁿⁿ} 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">(</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {A(\,·\,,y)} is uniformly Holder continuous and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mo>inf</m:mo> <m:mrow> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> </m:msub> <m:mo>⁡</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> <m:mo>></m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {inf_{xⁿ}A(x,x)>0} , then for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>L</m:mi> <m:mi>loc</m:mi> <m:mi>p</m:mi> </m:msubsup> </m:mrow> </m:math> {f∈ Lᵖloc} , for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {p≥ 2} , the solution vector <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>u</m:mi> <m:mo>∈</m:mo> <m:msubsup> <m:mi>H</m:mi> <m:mi>loc</m:mi> <m:mrow> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo>-</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>p</m:mi> </m:mrow> </m:msubsup> </m:mrow> </m:math> {u∈ H2s-δ,ploc} for some <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>δ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {δ∈(0,s)} .