SHARP WEAK-TYPE ESTIMATES FOR THE DYADIC-LIKE MAXIMAL OPERATORS

Abstract

We provide the explicit formula for the Bellman function corresponding to the weak-type $L^{p,\infty}\to L^{q,\infty}$ estimates for the dyadic maximal operator on $\mathbb{R}^n$. Actually, we do this in the more general setting of maximal operators associated with a tree-like structure on a nonatomic probability space. The approach rests on a clever combination of some novel optimization and combinatorial arguments.