Nonlocal Operators with singular anisotropic kernels

We study nonlocal operators that generate anisotropic jump processes, such as a jump process that behaves like a stable process in each direction but with a different index of stability. Its generator is the sum of one-dimensional fractional Laplace operators with different orders of differentiability. We study such operators in the general framework of bounded measurable coefficients. 
The objective of this talk is to provide regularity results for weak solutions to corresponding integro-differential equations.  
Joint work with Moritz Kassmann.