On a family of fully nonlinear integrodifferential operators : from fractional Laplacian to nonlocal Monge–Ampère

Abstract

We introduce a new family of intermediate operators between the fractional Laplacian and the nonlocal Monge–Ampère introduced by Caffarelli and Silvestre that are given by infimums of integrodifferential operators. Using rearrangement techniques, we obtain representation formulas and give a connection to optimal transport. Finally, we consider a global Poisson problem prescribing data at infinity, and prove existence, uniqueness, and $C^{1,1}$-regularity of solutions in the full space.