Sparse bounds for maximally truncated oscillatory singular integrals
Bibliographic record
Abstract
For polynomial P(x, y), and any Calderón-Zygmund kernel, K, the operator below satisfies a (1, r) sparse bound, for 1 < r ≤ 2.The implied bound depends upon P(x, y) only through the degree of P. We derive from this a range of weighted inequalities, including weak type inequalities on L 1 (w), which are new, even in the unweighted case.The unweighted weak-type estimate, without maximal truncations, is due to Chanillo and Christ (1987).Theorem A. Under the assumptions above, there holds for a finite constant sup deg(P)=dThe L 1 theory is more delicate, with the dominant result being that of Chanillo and Christ [2] proving that the operators T P indeed map L 1 into weak L 1 , with again the bound depending only on the degree of P, the dimension, and the kernel K. Their argument does not address maximal truncations.Our main result proves sparse bounds for the maximal truncations
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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 | 0.018 |
| 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 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.005 | 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".