Properties of extremal sequences for the Bellman function of the dyadic maximal operator

We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator.This implies the weak-Lp uniqueness for such a sequence.Comment: 11 page

Extremal sequences for the Bellman function of the dyadic maximal operator

We give a characterization of the extremal sequences for the Bellman function of the dyadic maximal operator.In fact we prove that they behave approximately like eigenfunctions of this operator for a specific eigenvalue.Comment: 20 page

The Bellman function of the dyadic maximal operator in connection with the Dyadic Carleson Imbedding Theorem and related inequalities

We provide an alternative proof of the expression of the Bellman function of the dyadic maximal operator in connection with the Dyadic Carleson Imbedding Theorem, which appears in [10]. We also state and prove a sharp integral inequality for this operator in connection with the above Bellman function, and give an application

A sharp integral inequality for the dyadic maximal operator and a related stability result

We prove a sharp integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables is possible, as can be seen in [3]. Our inequality of interest is proved in this article by a simpler and more immediate way. We also study a stability result in connection with this inequality, that is we provide a necessary and sufficient condition, for a sequence of functions,under which we obtain equality in the limit. The proof of this result is based on the proof of the related inequality which we present in this article.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1502.06002, arXiv:1604.0278