Bibliographic record
Abstract
The main goal of this paper is to prove an optimal limiting Sobolev inequality in two dimensions for Hölder continuous functions. Additionally, from this inequality we derive the double logarithmic inequality <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar u double-vertical-bar Subscript upper L Sub Superscript normal infinity Baseline less-than-or-slanted-equals StartFraction double-vertical-bar nabla u double-vertical-bar Subscript upper L squared Baseline Over StartRoot 2 pi alpha EndRoot EndFraction NestedTwiceStartRoot ln left-parenthesis 1 plus 6 StartRoot 2 pi alpha EndRoot StartFraction double-vertical-bar u double-vertical-bar Subscript ModifyingAbove normal upper C With dot Sub Superscript alpha Subscript Baseline Over double-vertical-bar nabla u double-vertical-bar Subscript upper L squared Baseline EndFraction NestedStartRoot ln left-parenthesis 1 plus StartRoot 2 pi alpha EndRoot StartFraction double-vertical-bar u double-vertical-bar Subscript ModifyingAbove normal upper C With dot Sub Superscript alpha Subscript Baseline Over double-vertical-bar nabla u double-vertical-bar Subscript upper L squared Baseline EndFraction right-parenthesis NestedEndRoot right-parenthesis NestedTwiceEndRoot"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> <mml:mo> ⩽ </mml:mo> <mml:mfrac> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:msqrt> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi> α </mml:mi> </mml:msqrt> </mml:mfrac> <mml:msqrt> <mml:mi>ln</mml:mi> <mml:mo> </mml:mo> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.623em" minsize="1.623em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> <mml:mi> π </mml:mi> <mml:mi> α </mml:mi> </mml:msqrt> <mml:mstyle displaystyle="false" scriptlevel="0"> <mml:mfrac> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>u</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:msup> <mml:mrow> <mml:mo fence="true" stretchy="true" symmetric="true"/> <mml:mspace width="negativethinmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mi mathvariant="normal">C</mml:mi> <mml:mo> ˙ </mml:mo> </mml:mover> </mml:mrow> <mml:mo fence="true" stretchy="true" symmetric="true"/> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mspace width="negativethinmathspace"/> <mml:mi> α </mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi mathvariant="normal"> ∇ </mml:mi> <mml:mi>u</mml:mi> <mml:msub> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> </mml:mfrac> </mml:mstyle> <mml:msqrt> <mml:mi>ln</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:msqrt> <mml:mn>2</mml:mn> <mml:mi> π
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 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 | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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".