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

Abstract

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 $u\le0$ in $\mathbb R^n\setminus B_1$, where $δ_yu(x)=u(x+y)+u(x-y)-2u(x)$, 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 $n\ge1$, $0<σ<2$, and $0<λ<Λ$, there exist a fixed coefficient $a(x,y)$ satisfying \textup{(C)}, a constant $d>0$, and nonnegative functions $u_j,f_j\in C_c^\infty(B_1)$ such that $L_au_j\ge-f_j$ in $B_1$, $0\le f_j\le1$, and $|\operatorname{supp}f_j|\to0$ as $j\to\infty$, while $u_j\ge d$ on $B_{1/4}$ for every $j$. Consequently, this fixed operator admits no $L^p$ maximum estimate for any $1\le p<\infty$.