Bellman function sitting on a tree

Hardy不等式在二元树上的Bellman函数

Bellman function for Hardy's inequality over dyadic trees

本文利用Bellman函数技术刻画了二元树上加权Hardy不等式成立的测度，证明了相关常数的最优性，并给出了Bellman函数的随机最优控制解释。

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.

Local lower norm estimates for dyadic maximal operators and related Bellman functions

We provide lower $L^q$ and weak $L^p$-bounds for the localized dyadic maximal operator on $R^n$, when the local $L^1$ and the local $L^p$ norm of the function are given. We actually do that in the more general context of homo- geneous tree-like families in probability spaces.

Inequalities for BMO on $α$-trees

We develop technical tools that enable the use of Bellman functions for BMO defined on $α$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $α$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be…

The ${\rm BMO}\to{\rm BLO}$ action of the maximal operator on $α$-trees

We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural operator equals 1 for all $n,$ and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called $α$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $α,$ but also allows a majorant independent of $α,$ hence…

Bi-parameter embedding and measures with restriction energy condition

Nicola Arcozzi, Pavel Mozolyako, Karl-Mikael Perfekt, and Giulia Sarfatti recently gave the proof of a bi-parameter Carleson embedding theorem. Their proof uses heavily the notion of capacity on bi-tree. In this note we give one more proof of a bi-parameter Carleson embedding theorem that avoids the use of bi-tree capacity. Unlike the proof on a simple tree (in a pervious paper of the authors) that used the Bellman function technique, the proof here is based on some rather subtle comparison of energies of measures on bi-tree.