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 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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.004 | 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".