Nonlocal equations, often modeled using the fractional Laplacian, have received significant attention in recent years. In this talk, we will briefly overview how the classical regularity theory for second-order (elliptic) PDEs (in divergence form) has been extended to fractional-order nonlocal equations. We will explore the Schauder, De Giorgi–Nash–Moser, Morrey–Campanato, and Calderón–Zygmund theories, and present some open problems in these fields.
