The hunt for a Bellman Function.
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 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- version of the Carleson Imbedding Theorem
Let be the set of all dyadic intervals of the real line. Given a function , consider the averages , on each dyadic interval .
Let be a family of non-negative real values satisfying the Carleson measure condition—that is, for any dyadic interval ,
Then, there is a constant such that for any ,
Fix a dyadic interval , and a vector . Consider all families satisfying the Carleson condition
and such that
Also, consider all functions for which the following quantities are fixed:
If we believe that the Theorem is true, then the quantity
is finite and, moreover, satisfies the inequality .
Since does not depend on the choice of an interval , we obtain a function of three real variables; this is the Bellman function associated with the Carleson Imbedding Theorem.
- The domain of is the set
- For each in the domain of , it is
- If , then
whenever the triples , and belong to the domain and
The entire machine can be run backward: if we have any function of three real variables that satisfies properties 1—3, the proof of the Theorem follows immediately. The key property 3 is not very pleasant to verify. Fortunately, this condition can be replaced by “infinitesimal” conditions (conditions on derivatives), which are easier to check: If , and , and all triples are in the domain of , then the key property 3 implies the concavity of :
Notice that condition 3 is equivalent to (eq3). The following function satisfies 1, 2 and (eq3), and thus the Theorem is proven for .