Aleksandrov–Bakelman–Pucci Type Estimates for Integro-Differential Equations

Abstract

In this work we provide an Aleksandrov–Bakelman–Pucci type estimate for a certain class of fully nonlinear elliptic integro-differential equations, the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal, fractional-order setting and on the use of Riesz potentials to interpret second derivatives as fractional order operators. This result applies to a family of equations involving some nondegenerate kernels and, as a consequence, provides some new regularity results for previously untreated equations. Furthermore, this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the L ∞ and L n norms of the right-hand side, in contrast to previous comparison results which utilize the continuity of the right-hand side for their conclusions. These results appear to be new, even for the linear case of the relevant equations.