Maz'ya's Question on Distinguishing Two Maximal Operators

Precise statement

Maz'ya and Shaposhnikova introduced a non-classical maximal operator $M^\diamond$, the maximal convolution with the vector-valued signum kernel truncated to centered balls. One of Maz'ya's 75 open problems in analysis asks whether it can be separated from the sharp maximal operator $M^\sharp$. It can: there is a translation-invariant Banach space of locally integrable functions on which $M^\diamond$ is bounded but $M^\sharp$ is not.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

ChatGPT 5.4 Pro and ChatGPT 5.5 Pro

Instructive in both directions. ChatGPT 5.4 Pro helped rule out candidate examples, specifically the classical function spaces already covered in the literature, which is search-space pruning rather than construction. Then after the author had finalized the proof, ChatGPT 5.5 Pro produced an alternative example the author calls surprisingly simple but not entirely legitimate, and the paper prints it anyway. The formal declaration is unambiguous: the ideas, results, proofs, bibliography and writing are entirely the author's work.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

The author states the mathematics is entirely his own and the models were used to investigate candidate examples. arXiv preprint, not peer-reviewed.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-05-17

   arXiv:2605.17663 - A Banach space that distinguishes two maximal operators

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

construction (source-reported)

Source-reported tools: construction.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.