A counterexample to integral Aleksandrov-Bakelman-Pucci estimates for a nonlocal operator

We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form \begin{equation*} L_au(x):=\int_{\mathbb R^n}δ_yu(x) \frac{a(x,y)}{|y|^{n+σ}},dy \ge-f(x)\quad$\text{in }$B_1, \end{equation*} with $u\le 0$ in ${\mathbb{R}}^n\setminus B_1$, where ${\delta }_{y}u(x)=u(x+y)+u(x-y)-2u(x)$, and the measurable coefficient satisfies \begin{equation}\tag{C} λ\le a(x,y)\leΛ,\qquad a(x,y)=a(x,-y),\qquad x\in\mathbb R^n,\quad y\ne0. \end{equation} More precisely, for every $n\ge 1$, $0<\sigma <2$, and $0<\lambda <\mathrm{\Lambda }$, there exist a fixed coefficient $a(x,y)$ satisfying \textup{©}, a constant $d>0$, and nonnegative functions $u_j,f_j\in C_c^{\mathrm{\infty }}(B_1)$ such that $L_a{u}_j\ge -f_j$ in $B_1$, $0\le f_j\le 1$, and $\mathrm{\mid }\mathrm{supp}{f}_j\mathrm{\mid }\to 0$ as $j\to \mathrm{\infty }$, while $u_j\ge d$ on $B_{1\mathrm{/}4}$ for every $j$. Consequently, this fixed operator admits no $L^p$ maximum estimate for any $1\le p<\mathrm{\infty }$.