Bessel potentials and optimal Hardy and Hardy-Rellich inequalities
Bibliographic record
Abstract
We give necessary and sufficient conditions on a pair of positive radial functions V and W on a ball B of radius R in R^n,$n \geq 1$, so that the following inequalities hold for all $u \in C_{0}^{\infty}(B)$: $\int_{B}V(x)|\nabla u |^{2}dx \geq \int_{B} W(x)u^2dx$, and $\int_{B}V(x)|Δu |^{2}dx \geq \int_{B} W(x)|\nabla u|^{2}dx+(n-1)\int_{B}(\frac{V(x)}{|x|^2}-\frac{V_r(|x|)}{|x|})|\nabla u|^2dx$. This characterization makes a very useful connection between Hardy-type inequalities and the oscillatory behaviour of certain ordinary differential equations, and helps in the identification of a large number of such couples (V, W) - that we call Bessel pairs -as well as the best constants in the corresponding inequalities. This allows us to improve, extend, and unify many results -old and new- about Hardy and Hardy-Rellich type inequalities, such as those obtained by Caffarelli-Kohn-Nirenberg, Brezis-Vazquez, Wang-Willem, Adimurthi-Chaudhuri-Ramaswamy, Filippas-Tertikas, Adimurthi-Grossi -Santra, Tertikas-Zographopoulos, and Blanchet-Bonforte-Dolbeault-Grillo-Vasquez.
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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 | 0.003 | 0.010 |
| 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 | 0.004 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 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".