A two weight theorem for $\alpha$-fractional singular integrals with an energy side condition
Bibliographic record
Abstract
Let \sigma and \omega be locally finite positive Borel measures on \mathbb{R}^{n} with no common point masses, and let T^{\alpha} be a standard \alpha -fractional Calderón–Zygmund operator on \mathbb{R}^{n} with 0 \leq \alpha < n . Furthermore, assume as side conditions the \mathcal{A}_{2}^{\alpha} conditions and certain \alpha -energy conditions . Then we show that T^{\alpha} is bounded from L^{2}(\sigma ) to L^{2}( \omega ) if the cube testing conditions hold for T^{\alpha} and its dual, and if the weak boundedness property holds for T^{\alpha} . Conversely, if T^{\alpha} is bounded from L^{2}( \sigma ) to L^{2}( \omega ) , then the testing conditions hold, and the weak boundedness condition holds. If the vector of \alpha -fractional Riesz transforms \mathbf{R}_{\sigma }^{\alpha} (or more generally a strongly elliptic vector of transforms) is bounded from L^{2}( \sigma) to L^{2}( \omega ) , then the \mathcal{A}_{2}^{\alpha} conditions hold. We do not know if our energy conditions are necessary when n \geq 2 . The innovations in this higher dimensional setting are the control of functional energy by energy modulo \mathcal{A}_{2}^{\alpha} , the necessity of the \mathcal{A}_{2}^{\alpha} conditions for elliptic vectors, the extension of certain one-dimensional arguments to higher dimensions in light of the differing Poisson integrals used in \mathcal A_2 and energy conditions, and the treatment of certain complications arising from the Lacey–Wick monotonicity lemma. The main obstacle in higher dimensions is thus identified as the pair of energy conditions.
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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 | 0.002 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| 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".