Higher Hölder regularity for the fractional $p$-Laplace equation in the subquadratic case

We study the fractional $p$-Laplace equation $$(-{\mathrm{\Delta }}_p{)}^{s}u=0$$ for $0<s<1$ and in the subquadratic case $1<p<2$. We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp. Our results complement the previous results for the superquadratic case when $p\ge 2$. The arguments are based on a careful Moser-type iteration and a perturbation argument.