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A Banach space that distinguishes two maximal operators
Abstract
Maz'ya and Shaposhnikova introduced a non-classical maximal operator Mdiamond as the maximal convolution with the vector-valued signum kernel truncated to centered balls. We construct a translation-invariant Banach space of locally integrable functions on which Mdiamond is bounded, but the sharp maximal operator M♯ is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.
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BibTeX
@misc{kovac2026banach,
title = {{A Banach space that distinguishes two maximal operators}},
author = {Vjekoslav Kovač},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.17663},
eprint = {2605.17663},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.17663},
}
RIS
TY - UNPB
TI - A Banach space that distinguishes two maximal operators
AU - Vjekoslav Kovač
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2605.17663
UR - https://arxiv.org/abs/2605.17663
AB - Maz'ya and Shaposhnikova introduced a non-classical maximal operator M^diamond as the maximal convolution with the vector-valued signum kernel truncated to centered balls. We construct a translation-invariant Banach space of locally integrable functions on which M^diamond is bounded, but the sharp maximal operator M^♯ is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.
ER -