Higher H\"older regularity for the fractional $p-$Laplacian in the superquadratic case

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 $q>N\mathrm{/}(sp)$, which is almost sharp whenever $sp\le (p-1)+N\mathrm{/}q$. The result is new already for the homogeneous equation.