Abstract We introduce atoms for dyadic atomic $${\mathbb {H}}^1$$ H 1 which the equivalence between and maximal function definitions is dimension independent. give sharp, up to $$\log (d)$$ log ( d</mml:mi...

We investigate the Fefferman-Stein inequality related a function f and the sharp maximal function f on a Banach function space X. It is proved that this inequality is equivalent to a certain boundedness property of the Hardy-Littlewood maximal operatorM . The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spa...