**4. Conclusion**

This chapter is dedicated to studying of the fundamental solutions to the generalization of the fractional Laplaciane by the methods of the theory of perturbation. We establish the regularity properties of the solutions to the fractional Laplacian equation with perturbations of different types. The Harnack inequality of a weak solution in the Sobolev space to the fractional Laplacian problem is studied, and the oscillation of the solution to the fractional Laplacian is estimated.
