Conformally covariant boundary operators and sharp higher order CR Sobolev trace inequalities on the Siegel domain and complex ball
Bibliographic record
Abstract
Abstract We first introduce an appropriate family of conformally covariant boundary operators associated to the Siegel domain U n + 1 \mathcal{U}^{n+1} with the Heisenberg group H n \mathbb{H}^{n} as its boundary and the complex ball B C n + 1 \mathbb{B}_{\mathbb{C}}^{n+1} with the complex sphere S 2 n + 1 \mathbb{S}^{2n+1} as its boundary. We provide the explicit formulas of these conformally covariant boundary operators. Second, we establish all higher order extension theorems of Caffarelli–Silvestre type for the Siegel domain and complex ball. Third, we prove all higher order CR Sobolev trace inequalities for the Siegel domain U n + 1 \mathcal{U}^{n+1} and the complex ball B C n + 1 \mathbb{B}_{\mathbb{C}}^{n+1} . In particular, we generalize the Sobolev trace inequality in the CR setting by Frank–González–Monticelli–Tan in the case γ ∈ ( 0 , 1 ) \gamma\in(0,1) to the general case for all γ ∈ ( 0 , n + 1 ) ∖ N \gamma\in(0,n+1)\setminus\mathbb{N} . The family of higher order conformally covariant boundary operators we define is naturally intrinsic to the higher order Sobolev trace inequalities on both the Siegel domain U n + 1 \mathcal{U}^{n+1} and complex ball B C n + 1
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.004 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.007 | 0.001 |
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 source (direct Gemma or distilled Codex), 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".