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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">U</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathcal{U}^{n+1} with the Heisenberg group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">H</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> \mathbb{H}^{n} as its boundary and the complex ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">B</m:mi> <m:mi mathvariant="double-struck">C</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> \mathbb{B}_{\mathbb{C}}^{n+1} with the complex sphere <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="double-struck">S</m:mi> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>n</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">U</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathcal{U}^{n+1} and the complex ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">B</m:mi> <m:mi mathvariant="double-struck">C</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msubsup> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>γ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> \gamma\in(0,1) to the general case for all <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>γ</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>∖</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:mrow> </m:mrow> </m:math> \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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi mathvariant="script">U</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> \mathcal{U}^{n+1} and complex ball <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mi mathvariant="double-struck">B</m:mi> <m:mi mathvariant="double-struck">C</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow>
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.002 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.001 | 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".