Gradient Hölder regularity for singular fractional $p$-Laplace equations

Abstract

Let $n\ge 2$, $1<p<2$, $0<s<1$, and $sp>p-1$. We prove that every globally bounded fractional $p$-harmonic function is locally $C^{1,\alpha }$ for some $\alpha =\alpha (n,p,s)>0$. This settles the open problem of interior gradient Hölder regularity in the singular range throughout the natural first-order regime $sp>p-1$. The proof combines an affine-invariant improvement-of-flatness argument with a Liouville theorem for globally Lipschitz entire solutions. In the large-slope regime, the shifted Bregman energies converge to an anisotropic stable form of order $sp-p+2>1$. In the bounded-slope regime, the Liouville theorem follows from rigidity of extremal secants, a recurrent blow-down argument, and a directional Morrey-Kato estimate for the singular linearized kernel. An affine Campanato argument controls the variation of the best affine approximations across scales. These estimates yield a scale-invariant decay of the affine excess and hence the local $C^{1,\alpha }$ estimate.