COMPOSITION OF HAAR PARAPRODUCTS: THE RANDOM CASE
Bibliographic record
Abstract
Abstract. When is the composition of paraproducts bounded? This is an important, and difficult question. We consider randomized variants of this question, finding non-classical characterizations. For dyadic interval I, let hI = h0 I be the L2-normalized Haar function adapted to I, the superscript 0 denoting that it has integral zero. Set h1 I = |hI|, the superscript 1 denoting a non-zero integral. A (classical dyadic) paraproduct with symbol b is one of the operators B(b, f) = ∑ 〈b, hI〉 √ 〈f, h |I | ǫ I 〉hδI. I∈D Here, ǫ, δ ∈ {0, 1}, with one of the two being zero and the other one. We characterize when certain randomized compositions B(b, B(β, ·)) are bounded operators on L 2 (R), permitting in particular both paraproducts to be unbounded. 1. Definitions and Main Theorems We phrase the (difficult) open question which motivates the consideration of this paper. Let D be the dyadic grid, and {hI: I ∈ D} the L2 normalized Haar basis, namely hI = |I | −1/2 ( −1Ileft
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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 | 0.007 | 0.027 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.003 | 0.008 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.002 | 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".