Gradient regularity for (s, p) -harmonic functions

We study the local regularity properties of (s, p) -harmonic functions, i. e. local weak solutions to the fractional p-Laplace equation of order s (0, 1) in the case p (1, 2]. It is shown that (s, p) -harmonic functions are weakly differentiable and that the weak gradient is locally integrable to any power q 1. As a result, (s, p) -harmonic functions are H\"older continuous to arbitrary H\"older exponent in (0, 1). In addition, the weak gradient of (s, p) -harmonic functions has certain fractional differentiability. All estimates are stable when s reaches 1, and the known regularity properties of p-harmonic functions are formally recovered, in particular the local W^2, 2-estimate.