A WEIGHTED MAXIMAL WEAK‐TYPE INEQUALITY

Let w be a dyadic A p weight ( 1 ⩽ p < ∞ ), and let M D be the dyadic Hardy–Littlewood maximal function on R d . The paper contains the proof of the estimate w { x ∈ R d : M D f ( x ) > w ( x ) } ⩽ C p[ w ] A p∫ R d| f | d x , where the constant C p does not depend on the dimension d . Furthermore, the linear dependence on[ w ] A pis 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.