Sharp L^{p,infty }rightarrow L^q Estimates for the Dyadic-Like Maximal Operators

For each (1le q we study the sharp versions of the (L^{p,infty }rightarrow L^q) estimates for the dyadic maximal operator on (mathbb {R}^n) . Actually, this is done in the more general setting of maximal operators associated with a tree-like structure. The proof rests on a novel combination of the Bellman function technique and optimization arguments.