##
**The hunt for a Bellman function: Applications to estimates of singular integral operators and to other classical problems in harmonic analysis.**
*(English.
Russian original)*
Zbl 0873.42011

St. Petersbg. Math. J. 8, No. 5, 721-824 (1997); translation from Algebra Anal. 8, No. 5, 32-132 (1996).

Recently, S. Treil and A. Volberg [J. Funct. Anal. (to appear)] characterized the matrix weights \(W\) for which the Hilbert transform acts on
\[
L^2(W)=\Biggl\{f:\Biggl(\int^{+\infty}_{-\infty} (W(t)f(t),f(t))dt\Biggr)^{1/2}<\infty\Biggr\}
\]
(a matrix weight is a function with values in the set of positive definite selfadjoint operators on a finite-dimensional Hilbert space \(E\); in the above formula, \(f\) is \(E\)-valued). A similar problem for \(L^p(W)\) with \(p\neq 2\) has turned out to be highly more difficult. This problem is solved in the paper under review.

The factual outline of the content of this paper is as follows. The authors find a natural analog of the Muckenhoupt condition and show that it is necessary for the boundedness of the Hilbert transform on \(L^p(W)\). To prove the sufficiency, they first show that, under the same condition on the weight, the Haar system is a good (“strong”) unconditional basis in \(L^p(W)\). Then they analyze the matrix \((H\varphi_I,\psi_J)\) of the Hilbert transform (in fact, of an arbitrary standard Calderón-Zygmund operator) relative to a couple \(\{\varphi_I\}\), \(\{\psi_J\}\) of Haar-type bases. A random choice of such a couple ensures with positive probability that, roughly speaking, a “Schur-type test” is applicable to this matrix, yielding the boundedness in question (this idea of randomization is a true novelty in similar problems). At the end, it is shown also that the maximal function operator acts on \(L^p(W)\) under the same condition on the matrix weight.

However, the review would be incomplete without saying how all this is done. The most difficult part of the proof, i.e., the analysis of the basis properties of the Haar system, is carried out with the help of an entirely new method. In fact, this method is applicable in many other situations (new and classical) involving certain embedding-type theorems (the authors start their analysis with the dyadic Carleson embedding theorem and the boundedness of the dyadic maximal operator, both things in the usual (nonweighted) \(L^2\)). The main idea is in constructing a specific function of several variables (a Bellman function for a particular problem in question) that satisfies certain concavity conditions. Once such a function is guessed and the conditions are verified, the proof of the required embedding inequality is easy and standard.

The Bellman function constructed by the authors for their main problem is not so terribly complicated that the verification of the required concavity-type properties would really present a problem for anybody; at the same time, it is hardly imaginable that, having the final formula only, anybody would be able to understand how it could be guessed. Not willing to give an impression of a miraculous trick, the authors explain step-by-step, gradually passing to more and more refined problems, how they managed to invent this formula. The long story of “the hunt for a Bellman function”, with all its difficulties and false traces followed by unexpected simplifications, which involves both heuristic argument and exact calculations (the latter, however, never becoming unbearably dull), is a quite fascinating piece of mathematical reading.

The factual outline of the content of this paper is as follows. The authors find a natural analog of the Muckenhoupt condition and show that it is necessary for the boundedness of the Hilbert transform on \(L^p(W)\). To prove the sufficiency, they first show that, under the same condition on the weight, the Haar system is a good (“strong”) unconditional basis in \(L^p(W)\). Then they analyze the matrix \((H\varphi_I,\psi_J)\) of the Hilbert transform (in fact, of an arbitrary standard Calderón-Zygmund operator) relative to a couple \(\{\varphi_I\}\), \(\{\psi_J\}\) of Haar-type bases. A random choice of such a couple ensures with positive probability that, roughly speaking, a “Schur-type test” is applicable to this matrix, yielding the boundedness in question (this idea of randomization is a true novelty in similar problems). At the end, it is shown also that the maximal function operator acts on \(L^p(W)\) under the same condition on the matrix weight.

However, the review would be incomplete without saying how all this is done. The most difficult part of the proof, i.e., the analysis of the basis properties of the Haar system, is carried out with the help of an entirely new method. In fact, this method is applicable in many other situations (new and classical) involving certain embedding-type theorems (the authors start their analysis with the dyadic Carleson embedding theorem and the boundedness of the dyadic maximal operator, both things in the usual (nonweighted) \(L^2\)). The main idea is in constructing a specific function of several variables (a Bellman function for a particular problem in question) that satisfies certain concavity conditions. Once such a function is guessed and the conditions are verified, the proof of the required embedding inequality is easy and standard.

The Bellman function constructed by the authors for their main problem is not so terribly complicated that the verification of the required concavity-type properties would really present a problem for anybody; at the same time, it is hardly imaginable that, having the final formula only, anybody would be able to understand how it could be guessed. Not willing to give an impression of a miraculous trick, the authors explain step-by-step, gradually passing to more and more refined problems, how they managed to invent this formula. The long story of “the hunt for a Bellman function”, with all its difficulties and false traces followed by unexpected simplifications, which involves both heuristic argument and exact calculations (the latter, however, never becoming unbearably dull), is a quite fascinating piece of mathematical reading.

Reviewer: S.V.Kislyakov (St.Peterburg)

### MSC:

42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |

42B25 | Maximal functions, Littlewood-Paley theory |

42A50 | Conjugate functions, conjugate series, singular integrals |