Abstract. For each q < 1 we precisely evaluate the main Bellman functions associated with the behavior of dyadic maximal operators on R n on integrable functions. Actually we do that in the more general setting of tree-like maximal operators. These are related to and refine the corresponding Kolmogorov inequality, which we show is actually sharp. For this we use the effective linearization introduced by the first author in 2005 for such maximal operators on an adequate set of functions.

We prove a sharp integral inequality for the dyadic maximal operator and give as an application another proof for the computation of its Bellman function of three variables.

We obtain sharp estimates for the localized distribution function of Mφ, when φ belongs to L p,∞ where M is the dyadic maximal operator. We obtain these estimates given the L 1 and L q norm, q < p and certain weak-L p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator.As a consequence we prove that the inequalityis sharp allowing every possible value for the L 1 and the L q norm for a fixed q such that 1 < q < p, where || · ||p,∞ is the usual quasi norm on L p,∞ .

We prove a sharp integral inequality which connects the dyadic maximal operator with the Hardy operator. We also give some applications of this inequality.

We obtain sharp estimates for the localized distribution function of the dyadic maximal function M d φ, given the local L 1 norms of φ and of G • φ where G is a convex increasing function such that G(x)/x → +∞ as x → +∞. Using this we obtain sharp refined weak type estimates for the dyadic maximal operator.