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

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