Sharp Lorentz estimates for dyadic-like maximal operators and related Bellman functions

We precisely evaluate Bellman type functions for the dyadic maximal opeator on $R^n$ and of maximal operators on martingales related to local Lorentz type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function $\varphi$ when the integral of $\varphi$ is fixed and also the same Lorentz quasinorm of $\varphi$ is fixed. Also we find the corresponding supremum when the integral of $\varphi$ is fixed and several weak type conditions are given.