Regularity for the fractional p-Laplace equation

Higher Sobolev and H\"older regularity is studied for local weak solutions of the fractional p-Laplace equation of order s in the case p 2. Depending on the regime considered, i. e. 0<s-2p or -2p<s<1, precise local estimates are proven. The relevant estimates are stable if the fractional order s reaches 1; the known Sobolev regularity estimates for the local p-Laplace are recovered. The case p=2 reproduces the almost W^1+s, 2 ₋₎₂-regularity for the fractional Laplace equation of any order s (0, 1).