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

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