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 Hölder continuous to arbitrary Hölder 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 W2,2-estimate.