Dyadic-like maximal operators on integrable functions and Bellman functions related to Kolmogorov’s inequality

Abstract

For each q > 1 q>1 we precisely evaluate the main Bellman functions associated with the behavior of dyadic maximal operators on R n \mathbb {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.