On the sharp multi-bubble stability for fractional Hardy-Sobolev equations -- A quantitative approach in low dimensions

We establish sharp quantitative multi-bubble stability for non-sign-changing critical points of the fractional Hardy-Sobolev inequality in the low-dimensional regime $2s<N<6s-2t$. For functions whose energy is close to that of a finite superposition of bubbles, we prove that the Euler-Lagrange deficit controls linearly the distance, in the homogeneous fractional Sobolev norm, to the multi-bubble manifold, and we recover the precise bubble configuration. This yields quantitative rigidity under arbitrary finite weak interactions. The proof combines a localization scheme adapted to the Hardy weight, weighted fractional Kato-Ponce commutator estimates, a bubble-wise spectral gap inequality, and a sharp interaction analysis. We also show that the linear rate is optimal by constructing a matching counterexample.