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Maz'ya's Question on Distinguishing Two Maximal Operators

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.

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analysisMay 17, 2026Significance 25/100Registry: unreviewed

Maz'ya's Question on Distinguishing Two Maximal Operators

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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…

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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.

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