A Banach space that distinguishes two maximal operators

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 $M^{\mathrm{♯}}$ is not. This answers one of Maz’ya’s questions from a collection of 75 open problems in analysis.