A dual Moser–Onofri inequality and its extensions to higher dimensional spheres
Bibliographic record
Abstract
We use optimal mass transport to provide a new proof and a dual formula to the Moser–Onofri inequality on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> . This is in the same spirit as the approach of Cordero-Erausquin, Nazaret and Villani [5] to the Sobolev and Gagliardo–Nirenberg inequalities and the one of Agueh–Ghoussoub–Kang [1] to more general settings. There are however many hurdles to overcome once a stereographic projection on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> is performed: Functions are not necessarily of compact support, hence boundary terms need to be evaluated. Moreover, the corresponding dual free energy of the reference probability density <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>μ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mi>π</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> is not finite on the whole space, which requires the introduction of a renormalized free energy into the dual formula. We also extend this duality to higher dimensions and establish an extension of the Onofri inequality to spheres <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>𝕊</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . What is remarkable is that the corresponding free energy is again given by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ρ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mi>ρ</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , which means that both the prescribed scalar curvature problem and the prescribed Gaussian curvature problem lead essentially to the same dual problem whose extremals are stationary solutions of the fast diffusion equations.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".