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 W2,2-estimate.