The Bellman functions of dyadic-like maximal operators and related inequalities

For each p>1 we precisely evaluate the main Bellman functions associated with the dyadic maximal operator on Rn and the dyadic Carleson imbedding theorem. Actually, we do that in the more general setting of tree-like maximal operators. These provide refinements of the sharp Lp inequalities for those operators. For this we introduce an effective linearization for such maximal operators on an adequate set of functions.