On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ 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 %on a boundary of a domain 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.