Explicit counterexamples to Frank-Loss optimizer conjectures for critical Dirac and vector Sobolev inequalities
Abstract
Frank and Loss proposed critical Sobolev inequalities for the Dirac operator and for vector fields. They conjectured optimality of their spinorial family and wrote that it is conceivable that the higher-dimensional Loss-Yau potential is an optimizer. Wang and Zhang recently disproved both optimizer claims using direct comparison and instability arguments. We compute closed-form quotients for two elementary families of explicit competitors with strictly smaller quotients. For every d≥2, the spinorial competitor is a constant spinor multiplied by a scalar Aubin-Talenti optimizer. For every d≥3, the vector competitor is the planar azimuthal field Ad(x)=d/((1+lvert xrvert2)2) (−x2,x1,0,…,0). It lies in the nonlinear gauge and its quotient is computed in closed form. For every odd d≥3, it gives an explicit proof of nonoptimality of the higher-dimensional Loss-Yau-Dunne-Min potential. In dimension three, this field is a specific member of the competitor family of Wang and Zhang: it is a linear combination of the Loss-Yau potential and its reflected twin. Its axisymmetry gives a negative answer to their question about minimization within the SO(2)× 1-invariant class.
Related
- Counterexamples to Two Conjectures on Tangles in Matroid Minors
- Non-freezing of the proximity digraph and non-pseudostable convergence in heterogeneous Hegselmann--Krause models: Counterexamples to conjectures and results of Mirtabatabaei and Bullo
- A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures
- Counterexamples to two conjectures on the diameter of clique-free graphs
- Cyclic-by-abelian counterexamples to the second and third Zassenhaus conjectures
BibTeX
@misc{bieganowski2026explicit,
title = {{Explicit counterexamples to Frank-Loss optimizer conjectures for critical Dirac and vector Sobolev inequalities}},
author = {Bartosz Bieganowski and Jarosław Mederski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.15787},
eprint = {2610.15787},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.15787},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Explicit counterexamples to Frank-Loss optimizer conjectures for critical Dirac and vector Sobolev inequalities
AU - Bartosz Bieganowski
AU - Jarosław Mederski
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.15787
UR - https://arxiv.org/abs/2610.15787
AB - Frank and Loss proposed critical Sobolev inequalities for the Dirac operator and for vector fields. They conjectured optimality of their spinorial family and wrote that it is conceivable that the higher-dimensional Loss-Yau potential is an optimizer. Wang and Zhang recently disproved both optimizer claims using direct comparison and instability arguments. We compute closed-form quotients for two elementary families of explicit competitors with strictly smaller quotients. For every d≥2, the spinorial competitor is a constant spinor multiplied by a scalar Aubin-Talenti optimizer. For every d≥3, the vector competitor is the planar azimuthal field A_d(x)=d/((1+lvert xrvert^2)^2) (−x_2,x_1,0,…,0). It lies in the nonlinear gauge and its quotient is computed in closed form. For every odd d≥3, it gives an explicit proof of nonoptimality of the higher-dimensional Loss-Yau-Dunne-Min potential. In dimension three, this field is a specific member of the competitor family of Wang and Zhang: it is a linear combination of the Loss-Yau potential and its reflected twin. Its axisymmetry gives a negative answer to their question about minimization within the SO(2)× 1-invariant class.
ER -