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.
Contributed talks by participants on related subjects will take place in the afternoon. Abstracts of proposals should be submitted to the organisers before March 15th 2011.(postponed to the end of April 2011)
Teachers : Alexander Volberg ( Michigan State University) et Vassily
Vasyunin (Institut Steklov
de Mathématiques - St.Petersbourg) |
