| Abstract |
Nonlocal operators are of significant interest in both analysis and probability theory. The thesis consists of four papers concerning interior and boundary regularity properties for nonlocal operators. The first and the second papers discuss the Krylov–Safonov theory and the Evans–Krylov and Schauder theories, respectively, for fully nonlinear nonlocal operators with rough kernels of variable orders. The interior regularity results, such as the Aleksandrov–Bakelman–Pucci estimates, Harnack inequality, H"older estimates, and generalized H"older estimates are established. The third paper studies the pointwise Green function estimates for a large class of nonlocal operators using purely analytic methods. In all three papers, the essence of the results is the robustness of the regularity estimates, which makes the theories for local and nonlocal operators unified.
On the other hand, the last paper deals with the boundary regularity estimates for nonlocal operators with kernels of variable orders. The nontrivial bahaviors of the solution to the Dirichlet problem near the boundary are captured by means of the renewal function. |
