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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 Lau(x):=∫ℝnδyu(x) a(x,y)/(|y|n+σ) dy ≥−f(x) in B1, with u≤0 in ℝn∖ B1, where δyu(x)=u(x+y)+u(x−y)−2u(x), and the measurable coefficient satisfies tagC λ≤ a(x,y)≤Λ, a(x,y)=a(x,−y), x∈ℝn, y≠0. More precisely, for every n≥1, 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 uj,fj∈ Cc∞(B1) such that Lauj≥−fj in B1, 0≤ fj≤1, and |suppfj|→0 as j→∞, while uj≥ d on B1/4 for every j. Consequently, this fixed operator admits no Lp maximum estimate for any 1≤ p<∞.
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BibTeX
@misc{jiang2026counterexample,
title = {{A counterexample to integral Aleksandrov-Bakelman-Pucci estimates for a nonlocal operator}},
author = {Yongsheng Jiang and Lei Xu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16162},
eprint = {2610.16162},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16162},
}
RIS
TY - UNPB
TI - A counterexample to integral Aleksandrov-Bakelman-Pucci estimates for a nonlocal operator
AU - Yongsheng Jiang
AU - Lei Xu
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16162
UR - https://arxiv.org/abs/2610.16162
AB - We construct a counterexample to integral Aleksandrov-Bakelman-Pucci estimates for nonlocal inequalities of the form L_au(x):=∫_ℝ^nδ_yu(x) a(x,y)/(|y|^n+σ) dy ≥−f(x) in B_1, with u≤0 in ℝ^n∖ B_1, where δ_yu(x)=u(x+y)+u(x−y)−2u(x), and the measurable coefficient satisfies tagC λ≤ a(x,y)≤Λ, a(x,y)=a(x,−y), x∈ℝ^n, y≠0. More precisely, for every n≥1, 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∈ C_c^∞(B_1) such that L_au_j≥−f_j in B_1, 0≤ f_j≤1, and |suppf_j|→0 as j→∞, while u_j≥ d on B_1/4 for every j. Consequently, this fixed operator admits no L^p maximum estimate for any 1≤ p<∞.
ER -