Optimality of the distance exponent in critical Hardy stability

Determine whether the exponent N of the distance functional in the quantitative stability estimate for the critical Hardy inequality is optimal.

Background

The paper establishes quantitative stability estimates for critical Hardy inequalities in which the distance from the family of virtual extremizers is raised to the exponent N. This improves the previously known exponent N2 obtained by Cianchi and Ferone. The exponent N arises from the dyadic summation argument used in the rearrangement-free proof, but the paper does not establish whether a larger exponent, a smaller exponent, or precisely N is the best possible choice.

Consequently, the unresolved issue is to characterize the sharp exponent governing the distance term in quantitative stability estimates for the critical Hardy inequality considered in the paper.

References

The exponent $N$ naturally arises from the dyadic summation argument in the proof and improves the previously known exponent $N{2}$. Whether the exponent $N$ is optimal remains an interesting open problem.

— Improved quantitative stability for the critical Hardy inequality  (2608.19732 - Sahu, 20 Aug 2026) in Remark following Section 4 (page number unavailable)