A finite-integrability ABP estimate for general nonlocal Pucci operators
Fix $n \ge 2$, $0 < s < 1$ and $0 < \lambda \le \Lambda$. With $\delta^2 u(x,y) = u(x+y) + u(x-y) - 2u(x)$, let $$\mathcal{M}^+ u(x) = \frac{1}{2} \int_{\mathbb{R}^n} \frac{\Lambda \bigl(\delta^2 u(x,y)\bigr)_+ - \lambda \bigl(\delta^2 u(x,y)\bigr)_-}{|y|^{n+2s}}\,dy$$ be the extremal operator of the linear operators with even kernels between $\lambda |y|^{-n-2s}$ and $\Lambda |y|^{-n-2s}$. Viscosity inequalities are tested with a smooth function near the contact point and the original $u$ outside that neighbourhood. For second-order uniformly elliptic equations the Aleksandrov–Bakelman–Pucci estimate bounds $\sup u$ by $C\|f\|_{L^n}$. For nonlocal equations such an estimate is known with $\|f\|_{L^\infty}$ (Caffarelli and Silvestre), and, with $p = n$ and $\vartheta = s$ in the inequality below, for operators with kernels of the form $y^{\mathsf T} A(x) y\, |y|^{-n-2s-2}$ (Guillen and Schwab).
Problem. Do there exist $1 \le p < \infty$, $\vartheta > 0$ and $C < \infty$, depending only on $n, s, \lambda, \Lambda$, such that every bounded upper semicontinuous viscosity subsolution of $$\mathcal{M}^+ u \ge f \ \text{ in } B_1, \qquad u \le 0 \ \text{ in } \mathbb{R}^n \setminus B_1,$$ with $f$ continuous and $\|f\|_\infty \le 1$, satisfies $$\sup_{B_1} u \le C \|f_-\|_{L^p(B_1)}^{\vartheta}?$$ The constants must not depend on the size of the support of $f$.
Source: Fernández-Real and Ros-Oton (2024), §3.6.2, Open question 3.3. Stated by AIM — Open Applied Problems with the extremal operator written out, the estimate in terms of f_-, and constants depending only on n, s, λ and Λ.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, Aleksandrov–Bakelman–Pucci estimates, integro-differential equations, Pucci extremal operators
References
@book{fernandezreal2024,
author = {Fern{\'a}ndez-Real, Xavier and Ros-Oton, Xavier},
title = {Integro-Differential Elliptic Equations},
series = {Progress in Mathematics},
volume = {350},
publisher = {Birkh{\"a}user},
address = {Cham},
year = {2024},
doi = {10.1007/978-3-031-54242-8},
eprint = {2411.12455},
archivePrefix = {arXiv},
}
@article{caffarelli2009,
author = {Caffarelli, Luis and Silvestre, Luis},
title = {Regularity theory for fully nonlinear integro-differential equations},
journal = {Comm. Pure Appl. Math.},
volume = {62},
number = {5},
year = {2009},
pages = {597--638},
doi = {10.1002/cpa.20274},
eprint = {0709.4681},
archivePrefix = {arXiv},
}
@article{guillen2012,
author = {Guillen, Nestor and Schwab, Russell W.},
title = {Aleksandrov--{B}akelman--{P}ucci type estimates for integro-differential equations},
journal = {Arch. Ration. Mech. Anal.},
volume = {206},
number = {1},
year = {2012},
pages = {111--157},
doi = {10.1007/s00205-012-0529-0},
eprint = {1101.0279},
archivePrefix = {arXiv},
}
@misc{aim-nonlocal-pucci-finite-integrability-abp,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{A finite-integrability ABP estimate for general nonlocal Pucci operators}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/462-nonlocal-pucci-finite-integrability-abp.md},
note = {Entry 462 at commit 24edaad, 2026-10-15},
}
Resolution claims
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 …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-18 | Needs verification → Solution claimed | — | Recorded here: arXiv:2610.16162v1. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00813, "A finite-integrability ABP estimate for general nonlocal Pucci operators", clopen.solutions.