Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains

Abstract: We prove a scale-invariant boundary Harnack inequality and Hölder continuity up to the boundary of quotients of positive fractional $p$-harmonic functions in Reifenberg flat domains with sufficiently small flatness constant, for $0<s<1$ and $2\le p&lt;2/(1-s)$. The solutions are globally nonnegative, vanish on a common exterior boundary patch, and belong to the natural local energy and $(p-1)$-tail class. Their distant exterior values may be unrelated and unbounded. After normalization at a corkscrew point, the comparison and Hölder constants depend only on $n,s,p$. We first establish local half-space boundary comparison for every $p\ge2$ and prove relative approximation by the half-space profile in Reifenberg domains. The compactness argument identifies escaped exterior mass and excludes its constant source term. Interior $C<sup>{1,α}$ regularity then yields gradient non-degeneracy and lower increment bounds in a cone of directions. To treat the degenerate kernels produced by linearization, we develop estimates for weighted non-local forms and prove a localized Harnack inequality for bounded quotients, retaining the full exterior interactions. Ordered variational deformations and a common fractional $p$-harmonic majorant yield comparison for solutions with unrelated exterior data. Truncation of distant data and fractional $p$-harmonic replacement extend the estimates from the variational energy class to the natural tail class.