Towards gradient Hölder regularity for singular fractional $p$-Laplace equations

Let $n\ge 2$, $1<p<2$, and let ${\alpha }_{\mathrm{l}\mathrm{o}\mathrm{c}}(n,p)$ be an admissible interior Hölder exponent for gradients of local $p$-harmonic functions. We prove that, for every $0<\alpha <{\alpha }_{\mathrm{l}\mathrm{o}\mathrm{c}}(n,p)$, there exists $s_{\ast }=s_{\ast }(n,p,\alpha )<1$ such that every bounded weak solution of $(-{\mathrm{\Delta }}_p{)}^{s}u=0$ in $B_2$ belongs to $C^{1,\alpha }(B_{1\mathrm{/}2})$ whenever $s\in (s_{\ast },1)$. The proof relies on an intrinsic excess-decay argument. We introduce a slope-normalized Bregman energy and establish compactness simultaneously in the bounded- and large-slope regimes. The corresponding blow-up limits are, respectively, minimizers of shifted local $p$-energies and solutions of uniformly elliptic constant-coefficient equations. A uniform improvement-of-flatness estimate for these two local families is then transferred to the fractional equation. The iteration is closed by separating the averaged exterior flux from the one-sided tails entering the upper and lower De Giorgi truncations, and by combining scale-invariant annular $L^p$ bounds with interpolation. This tail argument applies throughout the structural range $sp>p-1$, while the assumption that $s$ be close to one is used only in the compactness step. The result gives a partial answer to the open $C^{1,\alpha }$ problem in the singular range.