Bibliographic record
Abstract
We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathbb {R}^n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For the decreasing rearrangement <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f Superscript asterisk"> <mml:semantics> <mml:msup> <mml:mi>f</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:annotation encoding="application/x-tex">f^*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of a rearrangeable function <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f"> <mml:semantics> <mml:mi>f</mml:mi> <mml:annotation encoding="application/x-tex">f</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett–DeVore–Sharpley, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f Superscript asterisk Baseline double-vertical-bar Subscript upper B upper M upper O left-parenthesis double-struck upper R Sub Subscript plus Subscript right-parenthesis Baseline less-than-or-equal-to upper C Subscript n Baseline double-vertical-bar f double-vertical-bar Subscript upper B upper M upper O left-parenthesis double-struck upper R Sub Superscript n Subscript right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:msup> <mml:mi>f</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>BMO</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo> ≤ </mml:mo> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>f</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>BMO</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">\|f^*\|_{{\operatorname {BMO}}(\mathbb {R}_+)}\leq C_n \|f\|_{{\operatorname {BMO}}(\mathbb {R}^n)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , by showing the growth of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C Subscript n"> <mml:semantics> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">C_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in the dimension <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is not exponential but at most of the order of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartRoot n EndRoot"> <mml:semantics> <mml:msqrt> <mml:mi>n</mml:mi> </mml:msqrt> <mml:annotation encoding="application/x-tex">\sqrt {n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón–Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S f"> <mml:semantics> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>f</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">Sf</mml:annotation> </mml:semantics> </mml:math> </inline-formula> .
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 0.000 |
| Research integrity | 0.000 | 0.003 |
| Insufficient payload (model declined to judge) | 0.008 | 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".