September | 2008 | Francisco Blanco-Silva

The hunt for a Bellman Function.

September 26, 2008 1 comment

This is a beautiful and powerful mathematical technique in Harmonic Analysis that allows, among other things, to prove very complicated inequalities in the theory of Singular Integral Operators, without using much of the classical machinery in this field.

The Bellman function was the tool that allowed their creators (Fedor Nazarov and Sergei Treil) to crack the problem of weighted norm inequalities with matrix weights for the case $\boldsymbol{p} \neq \boldsymbol{2}$ and finally solve it completely.

Copies of the original paper can be found at the authors’ pages; e.g. [www.math.brown.edu/~treil/papers/bellman/bell3.ps] (notice the postscript file is huge, as the article has more than 100 pages).

Let me illustrate the use of Bellman functions to solve a simple problem:

Dyadic- $\boldsymbol{L}_\mathbf{2}(\mathbb{R})$ version of the Carleson Imbedding Theorem

Let $\mathcal{D}$ be the set of all dyadic intervals of the real line. Given a function $f \in L_1^{\text{loc}}(\mathbb{R})$ , consider the averages $\langle f \rangle_I = \lvert I\rvert^{-1} \int_I f$ , on each dyadic interval $I \in \mathcal{D}$ .
Let $\{ \mu_I \geq 0 \colon I \in \mathcal{D} \}$ be a family of non-negative real values satisfying the Carleson measure condition—that is, for any dyadic interval $I \in \mathcal{D}$ , $\sum_{J \subset I, J~\text{dyadic}} \mu_J \leq \lvert I \rvert.$
Then, there is a constant $C>0$ such that for any $f \in L_2(\mathbb{R})$ ,

$\displaystyle{\sum_{ I \in \mathcal{D} } \mu_I \lvert \langle f \rangle_{I} \rvert^2 \leq C \lVert f \rVert_{L_2(\mathbb{R})}^2}$