LAUSR

Let w be a dyadic Ap weight (1 ≤ p < ∞) and let MD be the dyadic Hardy-Littlewood maximal function on Rd. The paper contains the proof of the estimate w ({ x ∈ R : Mf(x) > w(x) }) ≤ Cp[w]Ap ∫ 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 <∞. 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.