A Signed-Weight Counterexample to a Grushin Hardy--Rellich Converse
Manuscript 22 October 2026 · Online 23 October 2026
Abstract
We construct a smooth signed radial weight for which a Grushin Hardy--Rellich inequality holds for every compactly supported smooth test function on the punctured unit gauge ball, but the associated Bessel equation has no positive solution. The principal weight is the constant one and the homogeneous dimension is five. The second weight is specified by the lowest eigenvalue of -d^2/dr^2+r^(-4) on (0,1). A quantitative spectral estimate on a codimension-one subspace yields the explicit positive margin 51/8900, and the Grushin spherical decomposition extends the inequality from radial tests to all tests. This is a counterexample to the remainder-free inequality with an unrestricted signed second weight, not to the forward Bessel-pair theorem or to nonnegative-weight variants. The preprint is unrefereed and makes no absolute priority claim.
Record
- Affiliation
- Mercury Software GmbH
- Contact
- [email protected] · GitHub
- Result
- Signed-weight counterexample; positive-weight scope not resolved
- Categories
- math.AP · math.CA
- Manuscript
- 22 October 2026
- Online release
- 23 October 2026
- Version
- 1.0
- License
- Creative Commons Attribution 4.0 International
Files and verification
The PDF is the canonical reading copy. The source archive contains the LaTeX manuscript, bibliography, reproducibility material, and audit documents without build artefacts.
Citation
Alper Ferudun, “A Signed-Weight Counterexample to a Grushin Hardy--Rellich Converse,” EulerSolve Research Papers, OWR-14299093-005, 2026. https://doi.org/10.5281/zenodo.23587424.
BibTeX
@misc{ferudun2026owr14299093005,
author = {Ferudun, Alper},
title = {A Signed-Weight Counterexample to a Grushin Hardy--Rellich Converse},
year = {2026},
howpublished = {EulerSolve Research Papers},
url = {https://eulersolve.org/papers/owr-14299093-005/},
doi = {10.5281/zenodo.23587424},
note = {OWR-14299093-005; unrefereed preprint}
}