Some Remarks on the Fefferman-stein Inequality

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 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(w), we obtain a new approach to the results of Sawyer and Yabuta concerning the Cp condition.