Maz'ya's Question on Distinguishing Two Maximal Operators
Statement
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.
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
construction · #1
Vjekoslav Kovač, using ChatGPT 5.4 Pro and ChatGPT 5.5 ProThat credit came with the record as it was imported. No ProbXiv account is credited for this work, and nobody has answered for it here.
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.
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