Sharp constants for the fractional Hardy inequalities with remainder

Determine the sharp constants in the fractional Hardy inequality with remainder and in the corresponding Sobolev–Bregman inequality with remainder on a half-space.

Background

The main theorems establish positive, explicit remainder constants for two nonlocal inequalities: a fractional Gagliardo-seminorm Hardy inequality and an inequality for Sobolev–Bregman forms. The constants are produced by ground-state representations and weighted fractional Hardy estimates, but the paper does not establish their optimality. The authors state that determining the sharp constants would require methods beyond those used in the paper.

References

The constants $C_{d, s, p, \tau}$ and $C'_{d, \alpha, p, \tau}$ are explicit, although of rather complicated form, and can be found in the proofs. However, there is no reason to believe that these constants are sharp.

— Fractional Hardy--Maz'ya inequality on a half-space  (2609.10832 - Kijaczko et al., 9 Sep 2026) in Section 1, subsection “Main results”