Sharp Constants and Optimizers for a Class of Caffarelli–Kohn–Nirenberg Inequalities
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Bibliographic record
Abstract
Abstract In this paper, we use a suitable transform of quasi-conformal mapping type to investigate the sharp constants and optimizers for the following Caffarelli–Kohn–Nirenberg inequalities for a large class of parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>r</m:mi> <m:mo>,</m:mo> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> <m:mo>,</m:mo> <m:mi>μ</m:mi> <m:mo>,</m:mo> <m:mi>σ</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(r,p,q,s,\mu,\sigma)} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>a</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0\leq a\leq 1} : <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>r</m:mi> </m:msup> <m:mfrac> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">|</m:mo> </m:mrow> <m:mi>s</m:mi> </m:msup> </m:mfrac> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mi>r</m:mi> </m:mfrac> </m:msup> <m:mo>≤</m:mo> <m:mi>C</m:mi> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mo>∇</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>p</m:mi> </m:msup> <m:mfrac> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>μ</m:mi> </m:msup> </m:mrow> </m:mfrac> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> <m:mfrac> <m:mi>a</m:mi> <m:mi>p</m:mi> </m:mfrac> </m:msup> <m:mrow> <m:mo maxsize="210%" minsize="210%">(</m:mo> <m:mo largeop="true" symmetric="true">∫</m:mo> <m:msup> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>u</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>q</m:mi> </m:msup> <m:mfrac> <m:mrow> <m:mi>d</m:mi> <m:mo></m:mo> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mo fence="true" stretchy="false">|</m:mo> <m:mi>x</m:mi> <m:msup> <m:mo stretchy="false">|</m:mo> <m:mi>σ</m:mi> </m:msup> </m:mrow> </m:mfrac> <m:mo maxsize="210%" minsize="210%">)</m:mo> </m:mrow> <m:mfrac> <m:mrow> <m:mn>1</m:mn> <m:mo>-</m:mo> <m:mi>a</m:mi> </m:mrow> <m:mi>q</m:mi> </m:mfrac> </m:msup> <m:mo>.</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> \bigg{(}\int\lvert u|^{r}\frac{dx}{|x|^{s}}\bigg{)}^{\frac{1}{r}}\leq C\bigg{(% }\int\lvert\nabla u|^{p}\frac{dx}{\lvert x|^{\mu}}\bigg{)}^{\frac{a}{p}}\bigg{% (}\int\lvert u|^{q}\frac{dx}{\lvert x|^{\sigma}}\bigg{)}^{\frac{1-a}{q}}. We compute the best constants and the explicit forms of the extremal functions in numerous cases. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>a</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {0<a<1} , we can de
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| 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 it