Bibliographic record
Abstract
This paper is devoted to presenting a new proof of boundedness of the commutator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b upper I Subscript alpha minus upper I Subscript alpha Baseline b"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:mo> − </mml:mo> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">bI_\alpha -I_\alpha b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript alpha"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b"> <mml:semantics> <mml:mi>b</mml:mi> <mml:annotation encoding="application/x-tex">b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are regarded as the Riesz and multiplication operators) acting on the Morrey space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Superscript p comma lamda"> <mml:semantics> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi> λ </mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">L^{p,\lambda }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> under <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b element-of upper B upper M upper O"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>BMO</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">b\in \operatorname {BMO}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and naturally, developing a regularity theory of commutators for Morrey-Sobolev spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript alpha Baseline left-parenthesis upper L Superscript p comma lamda Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mi> λ </mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I_\alpha (L^{p,\lambda })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> via a completely original iteration of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript alpha"> <mml:semantics> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">I_\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Even in the special case of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript alpha Baseline left-parenthesis upper L Superscript p Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi> α </mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">I_\alpha (L^p)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , this is a new theory.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.004 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.018 | 0.003 |
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".