$L^{p}$-Caffarelli-Kohn-Nirenberg inequalities and their stabilities
Bibliographic record
Abstract
We establish a general identity (Theorem 1.2) that implies both the $L^{p}$-Hardy identities and the $L^{p}$-Caffarelli-Kohn-Nirenberg identities (Theorems 1.3 and 1.4) and $L^{p}$-Hardy inequalities and the $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities (Theorems 1.5 and 1.6)). Weighted $L^{p}$-Caffarelli-Kohn-Nirenberg inequalities with nonradial weights are also obtained. (Theorem 1.7). Our results provide simple interpretations to the sharp constants, as well as the existence and non-existence of the optimizers, of several $L^{p}$-Hardy and $L^{p}% $-Caffarelli-Kohn-Nirenberg inequalities. As applications of our main results, we are able to establish stabilities of a class of $L^{2}$ and $L^{p}% $-Caffarelli-Kohn-Nirenberg inequalities. (Theorems 1.8 and 1.9.) We also derive the best constants and explicit extremal functions for a large family of $L^{2}$ and $L^{p}$ Caffarelli-Kohn-Nirenberg inequalities. (Corollaries 1.1 and 1.2.)
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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 | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".