A priori estimates for integro-differential operators with measurable kernels

The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators L of order alpha is an element of(0, 2). Thereby we extend the De Giorgi-Nash-Moser theory to non-local integro-differential operators. The operators L under consideration generate strong Markov processes via the theory of Dirichlet forms. As is well known, regularity properties of the resolvents are important for many aspects of the corresponding stochastic process. Therefore, this work is related to probability theory and analysis, especially partial differential equations, at the same time.