Higher Hölder regularity for the fractional $p-$Laplacian in the superquadratic case

Abstract: We prove higher H\"older regularity for solutions of equations involving the fractional $p-$Laplacian of order $s$, when $p\ge 2$ and $0<s\<1$. In particular, we provide an explicit H\"older exponent for solutions of the non-homogeneous equation with data in $L^q$ and $q>N/(s\,p)$, which is almost sharp whenever $s\,p\leq (p-1)+N/q$. The result is new already for the homogeneous equation.