Chapter 1 provides the necessary background and states the main results of the thesis. Two seperate topics are studied. The first topic is on two weight problems. The second topic is on Bellman functions on altered probability spaces. Chapter 2 proves a two weight estimation for a vector-valued positive operator. We consider two different cases of this theorem. The easier case 1 < p <= q requires only one testing condition. However, we construct a counterexample showing this testing condtion alone is not sufficient for the case q < p < 1. We apply the Rubio de Francia Algorithm to reduce our problem to the well-known two weight estimates for positive dyadic operators. Chapter 3 proves a two weight estimation for paraproducts. We again consider two different cases seperately. The first few steps of the proof proceeds exactly the same as in Chapter 2. However, for the harder case 2 < p < infinity, we need to characterize a two weight inequality for shifted bilinear forms, which takes up the majority of this chapter. Chapter 4 considers the celebrated Dyadic Carleson Embedding Theorem. We streamline a way of finding a super-solution of the Bellman function via the Burkholder's hull. We give an explicit formula of the Burkholder's hull and hence a super-solution in this chapter. Chapter 5 generalizes the Dyadic Carleson Embedding Theorem to the filtered probability spaces and proves the coincidence of the Bellman functions on an infinite refining filtration. The proof requires a remodeling of the Dyadic Carleson Embedding Theorem. Finally, we also consider the Bellman function of the Doob's Martingale Inequality.
Lai, Jingguo,
"Two weight problems and Bellman functions on filtered probability spaces"
(2015).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0WS8RM8
