\( H^{s, p}\) regularity theory for a class of nonlocal elliptic equations (Q2309308)

The very interesting paper under review extends the \(W^{1,p}\)-regularity theory of second-order, divergence form elliptic equations to the nonlocal setting. More precisely, given a domain \(\Omega\subset\mathbb{R}^n\), consider the nonlocal elliptic equation of the form \[ L_Au+bu=\sum_{i=1}^m L_{D_i}g_i+f, \] where \(b,g_i,f\colon\ \mathbb{R}^n\to\mathbb{R}\) are given functions and \[ L_Au(x)=2\lim_{\varepsilon\to0}\int_{\mathbb{R}^n\setminus B_\varepsilon(x)}\dfrac{A(x,y)}{|x-y|^{n+2s}}\big(u(x)-u(y)\big)\;dy,\quad x\in \Omega \] with \(s\in(0,1),\) \(n>2s.\) The kernels \(A,D_i\colon\ \mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}\) are measurable, symmetric and bounded, and \(A\) is supposed additionally to be elliptic and translation invariant. Setting \(H^{s,p}\) for the Bessel potential spaces, the main result of the paper asserts that if \(u\) is a weak solution of the nonlocal equation then \(u\in H^{s,2}\) implies \(u\in H^{s,p}\) for the whore range of exponents \(p\in(2,\infty).\) The approach in getting such nonlocal regularity relies on the technique of [\textit{L. A. Caffarelli} and \textit{I. Peral}, Commun. Pure Appl. Math., 51, No. 1, 1--21 (1998; Zbl 0906.35030)] based on comparison arguments, Vitali's covering lemma and estimates for the Hardy-Littlewood maximal function of the gradient, where, in the nonlocal settings the gradient of the weak solution is replaced by the \(s\)-gradient \(\nabla^su\), given by \[ \nabla^su(x)=\left(\int_{\mathbb{R}^n}\dfrac{\big(u(x)-u(y)\big)^2}{|x-y|^{n+2s}}\;dy\right)^{1/2}. \]