Optimal constant in the classical Hardy–Maz’ya inequality

Determine the optimal value of the constant C(p,τ) in the classical Hardy–Maz’ya inequality on a half-space, including the unresolved case p=2.

Background

The paper reviews the classical Hardy–Maz’ya inequality, which strengthens the Hardy inequality on a half-space by adding a weighted remainder involving the distance to a codimension-two subspace. Although positive lower bounds for the remainder constant C(p,τ) are known, the exact sharp value is not available. The authors emphasize that the difficulty already occurs in the local case, including p=2, where determining the constant is related to the first eigenvalue of a differential operator.

References

However, it is not known, what is the optimal value of $C(p,\tau)$, even for $p=2$ and the problem of finding the best constant in Hardyhalfspacewithremainder seems to be extremely challenging.

Hardyhalfspacewithremainder:

∫R∣∇u(x)∣p dx≥(p−1p)p∫R∣u(x)∣pxdp dx+C(p,τ)∫R∣u(x)∣pxdp−τ(xd−12+xd2)τ/2 dx,\int_{R}|\nabla u(x)|^p\,dx\ge\left(\frac{p-1}{p}\right)^p\int_{R}\frac{|u(x)|^p}{x_d^p}\,dx+C(p,\tau)\int_{R}\frac{|u(x)|^p}{x_d^{p-\tau}\left(x_{d-1}^2+x_d^2\right)^{\tau/2}}\,dx,

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

Moreover, it is not clear if the assumptions regarding the range of the parameter $\tau$ in Theorems \ref{thm1} and \ref{thm2} are optimal and we could not manage to answer this question in this paper.

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