Maz'ya's Question on Distinguishing Two Maximal Operators
Maz'ya and Shaposhnikova introduced a non-classical maximal operator , 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 . It can: there is a translation-invariant Banach space of locally integrable functions on which is bounded but is not.
- Result
- Proved
- Status
- Resolved
- AI contribution
- AI-assisted
- Method
- Construction
- Field
- Harmonic analysis
- Posed by
- Vladimir Maz'ya
- Year posed
- —
- Years open
- —
- Solved
- 2026-05-17
- Model
- ChatGPT 5.4 Pro and ChatGPT 5.5 Pro
- Vendor
- OpenAI
- Collaborators
- Vjekoslav Kovač
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 25 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What the AI did
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.
Verification
The author states the mathematics is entirely his own and the models were used to investigate candidate examples. arXiv preprint, not peer-reviewed.
Source
arXiv:2605.17663 - A Banach space that distinguishes two maximal operators