Bibliographic record
Abstract
Let ( X , M , μ ) be a σ -finite measure space and denote by P ( X ) the μ -measurable functions f : X → [ 0 , ∞ ] , f < ∞ μ ae. Suppose K : X × X → [ 0 , ∞ ) is μ × μ -measurable and define the mutually transposed operators T and T ′ on P ( X ) by ( T f ) ( x ) = ∫ X K ( x , y ) f ( y ) d μ ( y ) and ( T ′ g ) ( y ) = ∫ X K ( x , y ) g ( x ) d μ ( x ) , f , g ∈ P ( X ) , x , y ∈ X . Our interest is in inequalities involving a fixed (weight) function w ∈ P ( X ) and an index p ∈ ( 1 , ∞ ) such that: (*): ∫ X [ w ( x ) ( T f ) ( x ) ] p d μ ( x ) ≲ C ∫ X [ w ( y ) f ( y ) ] p d μ ( y ) . The constant C > 1 is to be independent of f ∈ P ( X ) . We wish to construct all w for which (*) holds. Considerations concerning Schur’s Lemma ensure that every such w is within constant multiples of expressions of the form ϕ 1 1 / p − 1 ϕ 2 1 / p , where ϕ 1 , ϕ 2 ∈ P ( X ) satisfy T ϕ 1 ≤ C 1 ϕ 1 and T ′ ϕ 2 ≤ C 2 ϕ 2 . Our fundamental result shows that the ϕ 1 and ϕ 2 above are within constant multiples of (**): ψ 1 + ∑ j = 1 ∞ E − j T ( j ) ψ 1 and ψ 2 + ∑ j = 1 ∞ E − j T ′ ( j ) ψ 2 respectively; here ψ 1 , ψ 2 ∈ P ( X ) , E > 1 and T ( j ) , T ′ ( j ) are the jth iterates of T and T ′ . This result is explored in the context of Poisson, Bessel and Gauss–Weierstrass means and of Hardy averaging operators. All but the Hardy averaging operators are defined through symmetric kernels K ( x , y ) = K ( y , x ) , so that T ′ = T . This means that only the first series in (**) needs to be studied.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".