A WEIGHTED MAXIMAL WEAK-TYPE INEQUALITY

Let w be a dyadic Ap weight (1p<<infinity>), and let MD be the dyadic Hardy-Littlewood maximal function on Rd. The paper contains the proof of the estimate w{x is an element of Rd:MDf(x)>w(x)}Cp[w]Ap integral Rd|f|dx, where the constant Cp does not depend on the dimension d. Furthermore, the linear dependence on [w]Ap is optimal, which is a novel result for 1<p<infinity. The estimate is shown to hold in a wider context of probability spaces equipped with an arbitrary tree-like structure. The proof rests on the Bellman function method: we construct an abstract special function satisfying certain size and concavity requirements.