The classical Aleksandrov–Bakel’man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue 2n-integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue n-integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate.

中文翻译：

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具有无限漂移的二阶椭圆算子的改进 Aleksandrov-Bakel'man-Pucci 估计

非散度形式的二阶椭圆算子的经典 Aleksandrov-Bakel'man-Pucci 估计（ABP 估计）是子解界的基本工具之一。Cabre 改进了 ABP 估计，将一个常数因子（给定域的直径）替换为几何特征，对于一些无界域，几何特征可以定义和有限。在证明中，Cabre 使用了来自 Trudinger 的 Krylov-Safonov 边界弱 Harnack 不等式；因此，要求一阶系数属于 Lebesgue2n- 可积函数空间。使用 Safonov 的增长引理和近似方法，我们将结果改进为 Lebesguen- 可积的一阶系数，它是最优的并且与原始 ABP 估计的条件一致。