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

Abstract

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.

Best constants in Muckenhoupt's inequality

“…See e.g. [4,5,6,7,8,13,14] and the monograph [3], consult also references therein. The primary goal of the present paper is to establish a sharp weighted version of (1.2).…”

Optimal weak type estimates for dyadic-like maximal operators

“…In fact it has been shown that This has been proved in a much more general setting of tree like maximal operators on non-atomic probability spaces. The result turns out to be independent of the choice of the measure space.The study of these operators has been continued in [7] where the Bellman functions of them in the case p < 1 have been computed. As in [5] and [7] we will follow the moregeneral approach.…”

“…The result turns out to be independent of the choice of the measure space.The study of these operators has been continued in [7] where the Bellman functions of them in the case p < 1 have been computed. As in [5] and [7] we will follow the moregeneral approach. So for a tree T on a non atomic probability measure space (X, µ), we define the associated dyadic maximal operator, namely…”

Extremal problems related to maximal dyadic-like operators

“…Such functions related to inequality (1.3) are precisely evaluated [1,2]. Actually if we define for any p > 1 (1.4) where Q is a fixed dyadic cube, φ is non-negative in L p (Q ) and F , f satisfy 0 f F 1/p .…”