Extremal problems related to maximal dyadic-like operators

Extremal sequences for the Bellman function of the dyadic maximal operator

“…Our aim in this article is to study this maximal operator and one way to do this is to find certain refinements of the inequalities satisfied by it such as (1.2) and (1.3). Concerning (1.2) refinements have been made in [8], [10] and [12]. Refinements of (1.3) can be found in [5] or even more general in [6].…”

The geometry of the dyadic maximal operator

“…An approach for studying the dyadic maximal operator is by making certain refinements of the above inequalities. Concerning (1.2), some of them have been done in [6], [10], [11], [12], while for (1.3) the Bellman function of this operator has been explicetely computed in [3]. It is defined by the following way: For every f, F, L such E-mail address: lefteris@math.uoc.gr that 0 < f p ≤ F , L ≥ f the Bellman function of three variables associated to the dyadic maximal operator is defined by:…”

“…for every 0 < f ≤ p p−1 F , k ∈ [0, 1] and 1 < q < p. Secondly it is known by [10] that the following inequality…”

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

“…It is easy to see that the weak type inequality (2) is the best possible. For refinements of this inequality see [7].…”

“…For the proof we use a symmetrization principle which appears in [6]. We also refine (7) to a another direction; more precisely we prove the following.…”