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We study the fractional $p$-Laplace equation $$ (-\Delta_p)^s u = 0 $$ for$0<s<1$ and in the subquadratic case $1<p<2$. We provide H\"older estimateswith an explicit H\"older exponent. The inhomogeneous equation is also treatedand there the exponent obtained is almost sharp. Our results complement theprevious results for the superquadratic case when $p\geq 2$. The arguments arebased on a careful Moser-type iteration and a perturbation argument.

Higher H\"older regularity for the fractional $p$-Laplace equation in the subquadratic case