LAUSR

Abstract In this paper, we systematically study the pointwise regularity for distribution solutions of fractional equations. We obtain a series of interior pointwise C k + [ 2 s + α ] , 2 s + α − [ 2 s + α ] regularities when 2 s + α is not an integer, C k + 2 s + α , ln ⁡ L regularities and C k + 2 s + α regularities when 2 s + α is an integer for k ≥ 0 under suitable conditions. These pointwise regularities seem to be more essential and characterize the solutions for fractional equations, and our proofs are more direct which can also provide optimal conditions for the above regularities. Furthermore, the classical regularity can be regarded as a consequence of pointwise regularity. The method we developed here can also be applied to many other equations, especially for the Poisson equations.