Sharp Homogeneous Weights in Fractional Hardy Inequalities

Definitions and mathematical statement

Write $x=(z,a)\in\mathbb H^n$, and let $\rho(x)=|(z,a)|$ denote the homogeneous norm intended in the source. Let $\mathcal L_s$ and $w_s$ denote, respectively, the source's fractional sub-Laplacian and its original Hardy weight. Their definitions and normalizations, the allowed values of $n$ and $s$, and the admissible function class are not supplied.

The intended conjecture is that replacing $w_s(z,a)$ by $\rho(z,a)^{2s}$ in the referenced sharp Hardy inequality preserves both the inequality and its sharp constant. The original passage refers to an "above inequality," but that inequality is absent from the supplied record. Consequently, even the placement of the weight cannot be recovered unconditionally.

For clarity, under the interpretation that the original inequality has the denominator-weight form \[ C_{n,s}\int_{\mathbb H^n}\frac{|f(x)|^2}{w_s(x)}\,\mathrm d\mu(x) \leq \langle\mathcal L_s f,f\rangle, \qquad f\in\mathcal D_{n,s}, \] the proposed replacement would assert \[ C_{n,s}\int_{\mathbb H^n}\frac{|f(x)|^2}{\rho(x)^{2s}}\,\mathrm d\mu(x) \leq \langle\mathcal L_s f,f\rangle, \qquad f\in\mathcal D_{n,s}, \] with $C_{n,s}$ still the largest constant valid for every admissible $f$. Here $\mathcal D_{n,s}$, $\mu$, the pairing, all parameter restrictions, and $C_{n,s}$ must be inherited from the missing original inequality; this display is a conditional formulation, not a reconstruction of those missing data. The treatment of the singularity at the identity and the finiteness requirements on the integrals likewise remain unspecified. A fully precise unconditional statement requires the omitted inequality and its definitions.

Short English statement

The supplied conjecture asks whether the original weight in a sharp fractional Hardy inequality on $\mathbb H^n$ can be replaced by $|(z,a)|^{2s}$ while preserving the same optimal constant. The preceding inequality, parameter ranges, operator and norm conventions, and admissible functions are missing, so the displayed formulation is conditional and requires mathematical review.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30005539$, source problem code catalogue entry, title "Sharp Homogeneous Weights in Fractional Hardy Inequalities," in the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2023.
• Supplied report citation: Hypoelliptic Operators in Geometry (2023), Oberwolfach Reports, pp. 1323--1396, DOI 10.4171/owr/2023/24. https://doi.org/10.4171/owr/2023/24. The supplied excerpt refers to an omitted preceding inequality.
• Supplied literature source, "OWR 2023 24": https://publications.mfo.de/bitstream/handle/mfo/4050/OWR_2023_24.pdf?isAllowed=y&sequence=4.
• Supplied literature source, "Roncal Thangavelu": https://arxiv.org/abs/1508.00714.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and states that the proposed replacement is not proved in the cited report or the established fractional Hardy literature. This is supplied dated curation, not an independently verified assessment; the record expressly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not independent verification in this draft.