The Bellman functions of the Carleson Embedding Theorem and the Doob's   martingale inequality
  Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work for the tree-like structure. In this paper, we give a self-complete proof of the coincidence of the Bellman functions on the more general infinitely refining filtered probability space, see Definition 1.3. The proof depends on a remodeling of the Bellman function of the dyadic Carleson Embedding Theorem. 
Jingguo Lai
math.CA math.PR