Uncentred maximal operators with respect to half balls on Damek--Ricci spaces

In this paper we study a variant of the uncentred Hardy–Littlewood maximal operator on Damek–Ricci spaces in which balls are replaced by suitable half balls. Perhaps surprisingly, such modified maximal operator has better boundedness properties than the classical one. In particular, it is bounded on $L^p$ for every $p$ in $(1,\mathrm{\infty }]$ (whereas the analogue operator on balls is bounded on $L^p$ only for $p>2$), and satisfies a limiting distributional inequality if $f$ is in $L\log (2+L)$. This endpoint estimate is optimal in the sense that it does not hold if $L\log (2+L)$ is replaced by a larger (in a suitable sense) Orlicz space.