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A priori estimates for integro-differential operators with measurable kernels. (English) Zbl 1158.35019
Summary: 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 $\alpha \in \left(0,2\right)$. 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.
##### MSC:
 35D10 Regularity of generalized solutions of PDE (MSC2000) 35B45 A priori estimates for solutions of PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35R05 PDEs with discontinuous coefficients or data 47G20 Integro-differential operators 60J75 Jump processes 45K05 Integro-partial differential equations
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