A quasiconformal composition problem for the $Q$-spaces
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Bibliographic record
Abstract
Given a quasiconformal mapping f:{\mathbb R}^n\to{\mathbb R}^n with n\ge2 , we show that (un-)boundedness of the composition operator {\mathbf C}_f on the spaces Q_{\alpha}({\mathbb R}^n) depends on the index \alpha and the degeneracy set of the Jacobian J_f . We establish sharp results in terms of the index \alpha and the local/global self-similar Minkowski dimension of the degeneracy set of J_f . This gives a solution to [3, Problem 8.4] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel–Lizorkin and Besov spaces. Consequently, Tukia–Väisälä's quasiconformal extension f:{\mathbb R}^n\to{\mathbb R}^n of an arbitrary quasisymmetric mapping g:{\mathbb R}^{n-p}\to {\mathbb R}^{n-p} is shown to preserve Q_{\alpha} ({\mathbb R}^n) for any (\alpha,p)\in (0,1)\times[2,n)\cup(0,1/2)\times\{1\} . Moreover, Q_{\alpha}({\mathbb R}^n) is shown to be invariant under inversions for all 0<\alpha<1 .
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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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 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 it