Sharper scale growth of corkscrew values

Prove a non-local counterpart of the scale-growth estimate C_ε^{-1}(ρ/R)^{s+ε} ≤ u(A_ρ(w))/u(A_R(w)) ≤ C_ε(ρ/R)^{s−ε} for 0 < ρ < R, under suitable control of the exterior data, thereby obtaining sharper growth estimates for positive fractional p-harmonic functions near the boundary.

Background

The main theorem supplies boundary comparison in sufficiently flat Reifenberg domains, but the concluding discussion asks whether the comparison estimates can be iterated to obtain a more precise power-law relation between values at corkscrew points on different scales.

Such an estimate would improve the available growth information and, as noted by the authors, a bound with exponent s+ε would remain integrable at infinity in the natural (p−1)-tail class when ε < s/(p−1). The question is explicitly left unresolved and is not needed for the proof of the main theorem.

References

Two further questions concern the present operator. First, removing the global domination 0 ≤F ≤ MqU from linear boundary comparison requires control of unbounded quotients and the exterior source in (6.24). Second, a non-local counterpart of [26, Lemma 3.15] would give (9.9) C−1ε( ρ R)s+ε≤ u(Aρ(w))u(AR(w)) ≤ Cε( ρ R)s−ε, 0 < ρ < R. An iteration of flat comparisons might yield this sharper growth, with suitable control of the exterior data. A power bound of order s+ε is integrable at infinity in the natural (p−1)-tail when ε < s/(p−1). Neither question is settled or used in Theorem 1.1.

— Boundary Harnack inequalities for the fractional $p$-Laplacian in Reifenberg flat domains  (2610.21689 - Nyström, 15 Oct 2026) in Equation (9.9) and Section 9.2, page 73