Global Weighted $L^2$ $\dbar$-solvability on Noncompact Pseudoconvex Complex Lie Groups
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Bibliographic record
Abstract
We prove global weighted $L^2$ solvability for the $\bar\partial$-equation on any noncompact pseudoconvex complex Lie group. If $G$ is a connected noncompact complex Lie group admitting a continuous plurisubharmonic (psh) exhaustion $\rho$, then for every $t \ge 0$, $p \ge 0$ and $q \ge 1$ the weighted $L^2$ Dolbeault cohomology $H^{p,q}_{\bar\partial,(2),t}(G)$ with respect to the weight $e^{-t\rho}$ vanishes and one has a global \emph{a priori} estimate. The argument relies on two uniformities provided by the Lie group geometry: (i) a uniform exhaustion by smoothly bounded strictly pseudoconvex domains whose defining functions approximate $\rho$ on fixed sublevels; (ii) strictly psh reference functions on these domains with a Levi lower bound independent of the exhaustion index. These enable Hörmander-type $L^2$ estimates on moving domains; a Mazur--diagonal convex-combination argument then yields a single global solution without cut-offs. Consequences include a Hartogs-type extension under weighted $L^2$ growth and richness of weighted Bergman spaces on strictly pseudoconvex sublevels.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
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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 it