In the present paper we prove estimates on subsolutions of the equation $-Av+c(x)v=0$, $x\in D$, where $D\subset {\mathbb{R}}^d$ is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered “quantitative” versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

在本文中，我们证明了方程的子解的估计 $-\mathrm{一种}v+C(X)v=0$, $X\in D$， 在哪里 $D\subset {\mathbb{电阻}}^d$是一个域（即开集和连通集），A是 Waldenfels 型的积分微分算子，其微分部分满足紧集上的一致椭圆度条件。一般而言，算子的系数不需要是连续的，而只需有界且可测量 Borel。我们的一些结果可能被认为是 Hopf 引理的“定量”版本，因为它们提供了在子解的最大值点处的外法向导数的下界，就其在该点的值而言。我们还将通过与A和域相关联的主本征函数显示围绕其最大值点的子解的下界。额外的结果，其中弱和强最大原理，弱 Harnack 不等式也得到了证明。