Dyadic-like maximal operators on LlogL functions

We study the following well-known property of the dyadic maximal operator Md on Rn (see [E.M. Stein, Note on the class LlogL, Studia Math. 32 (1969) 305–310] for the case of the Hardy–Littlewood maximal function): If ϕ is integrable and supported in a dyadic cube Q then Mdϕ is integrable over sets of finite measure if and only if |ϕ|log(1+|ϕ|) is integrable and the integral of Mdϕ can be estimated both from above and from below in terms of the integral of |ϕ|log(1+|ϕ|) over Q. Here we define and explicitly evaluate Bellman functions related to this property and the corresponding estimates (both upper and lower) for the integrals thus producing sharp improved versions of the behavior of Md on the local LlogL spaces.