Nonlinear nonlocal potential theory at the gradient level

The aim of this work is to establish numerous interrelated gradient estimates in the nonlinear nonlocal setting. First of all, we prove that weak solutions to a class of homogeneous nonlinear nonlocal equations of possibly arbitrarily low order have Hölder continuous gradients. Using these estimates in the homogeneous case, we then prove sharp higher differentiability as well as pointwise gradient potential estimates for nonlinear nonlocal equations of order larger than 1 in the presence of general measure data. Our pointwise estimates imply that the first-order regularity properties of such nonlinear nonlocal equations coincide with the sharp ones of the fractional Laplacian.