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The Hilbert transform-UMD dependence problem

Neither exponent can be lowered. Theorem 1.1 constructs explicit $2^{n}$ -dimensional spaces $X_n$ and $Y_n$ with $\hbar_{2,X_n}\asymp n$ , $\beta_{2,X_n}\asymp\sqrt n$ and $\beta_{2,Y_n}\asymp n$ , $\hbar_{2,Y_n}\asymp\sqrt n$ , with universal comparison constants, so both quadratic comparisons are sharp at $p=2$ ; Corollary 1.3 extends the growth rates to every fixed $1<p<\infty$ by extrapolation, with constants depending on $p$ . The spaces are built from the summation operators Wenzel had proposed as candidates for exactly this separation. The Lean formalisation covers Theorem 1.1 at $p=2$ over both real and complex scalars; the extrapolation to other $p$ and the exact coefficients of Remark 1.2 are outside it.

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 9, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-verified. Publication: preprint. AI contribution: ai-co-developed. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: The Hilbert transform-UMD dependence problem · Hilbert–UMD comparison

Confidence: Not scored

Registry verification: lean verified · preprint · resolved

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VibeMathed
registry · event recorded by

GPT-6 Astra
model · ai model contributor · OpenAI

Emiel Lorist
human · human collaborator

Jan van Neerven
human · human collaborator

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SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

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VibeMathed record: The Hilbert transform-UMD dependence problem

specialist registry · passed

Checked here on 12 September 2026 by reading the repository at commit 055d95e. Main/PaperStatement.lean states the theorem as the two dimensions and all eight inequalities, with $n$ and $\sqrt n$ literally as in the paper and universal constants. Mathlib has no UMD constant, so the two constants are project definitions and are the trust surface; both were read. umdConstant is the infimum of $C$ over all $\sigma$ -finite sample spaces, filtrations, $L^p$ martingales and unimodular coefficients for which the martingale transform is bounded by $C$ times the difference sum, and hilbertConstant is the infimum of $C$ over all $C^1$ compactly supported $f$ admitting a principal-value Hilbert transform in $L^p$ with norm at most $C\,\|f\|_p$ ; these are the standard notions. No sorry, no axiom declarations, no native_decide anywhere; tests/AxiomAudit.lean walks every project declaration transitively and fails on any admission or any axiom beyond propext, Classical.choice and Quot.sound. GitHub Actions ran the build and that audit on Ubuntu and Windows at the reviewed commit, both green, which puts the kernel check on machines other than the authors'. The build was not repeated here. Not formalised: Corollary 1.3 and the exact coefficients of Remark 1.2.