Primariness of Lp(L1)
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Does Lp(L1) have uniform primary factorization, and hence primariness, for every 1<p<∞? The target covers the full mixed-norm space, including its doubly cancellative product-Haar part.
Why it matters and what counts as progress
Why it matters: It asks for primariness and uniform primary factorization throughout the remaining mixed-norm Lp(L1) range.
What counts as progress: Prove primary factorization uniformly for every 1<p<∞, covering the mixed-norm and doubly cancellative components.
How the problem developed
Richard Lechner, Pavlos Motakis, Paul F. X. Müller and Thomas Schlumprecht Proved that L1(Lp) is primary for 1<p<∞; the later Lp(L1) problem concerns the reversed mixed-norm order.
- 19 Jul 2026 · Proof [2]
Antonio Acuaviva and Pablo Acuaviva, with AI-generated proof ideas Established the uniform primary factorization property, hence primariness, of Lp(L1) for every 1<p<∞. The paper says the model supplied the proof architecture and humans assembled and completed the written argument.
AI contribution
Mathematical Discovery in the Wild: AI-Guided Proofs in Banach Space Theory
For every 1<p<∞, Lp(L1) has uniform primary factorization and is therefore primary. The proof includes a scalar-compression argument on its doubly cancellative product-Haar part.
Sources
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