A Banach space that distinguishes two maximal operators

TL;DR

The paper constructs a specific Banach space where a non-classical maximal operator is bounded but the sharp maximal operator is not, addressing a question in analysis.

Contribution

It introduces a Banach space that distinguishes the boundedness properties of two maximal operators, solving an open problem posed by Maz'ya.

Findings

Constructed a Banach space where $M^ riangle$ is bounded.

Demonstrated $M^ riangle$ is bounded while $M^lat$ is not.

Answered a longstanding open problem in analysis.

Abstract

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 $M^\sharp$ is not. This answers one of Maz'ya's questions from a collection of 75 open problems in analysis.