Some remarks on the Fefferman-Stein inequality

We investigate the Fefferman-Stein inequality related to 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 operator M. The latter property is shown to be self-improving. We apply our results in several directions. First, we show the existence of nontrivial spaces X for which the lower operator norm of M is equal to 1. Second, in the case when X is the weighted Lebesgue space L(p) (w), we obtain a new approach to the results of Sawyer and Yabuta concerning the C(p) condition.