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In essence, the Bellman function method in Harmonic Analysis was introduced by Donald
Burkholder for finding the norm in Lp of the martingale transform. Apparently Burkholder
either did not notice or did not want to stress the link between his technique and Optimal
Control, but it becomes quite transparent if one carefully analyzes his proofs.
This unusual (for analysts) approach/insight of Burkholder can
be made into a method.
This method lays down a bridge between (stochastic) Optimal
Control and classical Harmonic Analysis, it allows to set up a clear
analogy between certain questions in Harmonic
Analysis and well studied methods in Optimal Control.
Soon it became clear that the scope of the method is quite wide,
in particular Nazarov-Treil-Volberg obtained by this method a necessary
and sufficient condition for the two weight
martingale transform to be bounded. This was used later in
Bellman estimates of Ahlfors-Beurling transform by Petermichl and
Volberg, which in turn did answer an old problem in
regularity theory of elliptic PDE.
Over last years, the Bellman function technique can be credited
for helping to solve several old Harmonic Analysis problems while
proposing a unified approach to many others.
In the first category one could name the sharp weighted
estimates of such classical operators as the Hilbert and Riesz
transforms (Petermichl) and the Ahlfors-Beurling transform
(Petermichl-Volberg). It was used also by Nazarov-Volberg to
solve a problem of Cohn
about embedding of the model space Kµ into L2(µ). Recently it
was shown that the Bellman function method can be used to solve another
famous Harmonic Analysis problem: the
so-called A2 conjecture on sharp weighted estimates of arbitrary
Calderon Zygmund operators. In the second category one can name all
kind of dimension free estimates of weighted
and unweighted Riesz transforms. Roughly speaking, the Bellman
function method makes
apparent the hidden multiscale properties of Harmonic Analysis
problems. It often replaces
sophisticated stopping time arguments of combinatorial nature by
relatively simple cookbook
recipes. Conversely, given a Harmonic Analysis problem with
certain scaling properties, one
can associate with it a non-linear PDE, the so-called stochastic
Bellman equation of the
problem.
Untill recently, in each case it was a matter of luck, experience and tenacity to nd a
supersolution , or even better a solution of this PDE (i.e. the Bellman function of the
problem). Recently, a method to nd such a solution for a whole class of problems has
emerged, largely due to the e orts of Vasyunin. So, the Bellman function technique is now
turning into a theory which can be learned and applied to a relatively wide range of issues.
The number of people interested in this method grows. Terry Tao has a couple of notes,
where he uses the method of Bellman function. In October 2010 there was a Fall School
devoted to applications of Bellman functions in Harmonic Analysis held at Lake Arrowhead,
CA. This school was organized by Christoph Thiele, Ignacio Uriarte Tuero and Alexander
Volberg. It gathered approximately 18 young researchers. It was be followed by a special
1
2 A. VOLBERG
session at an AMS meeting at UCLA. This area is very suitable for the young researchers,
because
a) it shows how methods from one seemingly distant eld (Optimal Control) can be used
in Analysis,
b) it generate progress in both elds,
c) it allows to quickly develop a working knowledge of some of the most powerful tools in
Analysis.
We plan to have approximately 8 lectures (by Alexander Volberg) mixed with recitations
(by Vasily Vasyunin), where the participants will be taught \hands-on" to build Bellman
functions.
Teachers : Alexander Volberg ( Michigan State University) et Vassily
Vasyunin (Institut Steklov
de Mathématiques - St.Petersbourg)
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