The Bellman functions for a certain two-weight inequality: A case study

Authors:
V. Vasyunin and A. Volberg

Translated by:
the authors

Original publication:
Algebra i Analiz, tom **18** (2006), nomer 2.

Journal:
St. Petersburg Math. J. **18** (2007), 201-222

MSC (2000):
Primary 42B20, 42A50, 47B35

Published electronically:
March 20, 2007

MathSciNet review:
2244935

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A formula is presented for the exact Bellman function of a certain ``toy'' two-weight problem. This adds one more function to a short list of other Bellman functions for which the precise expressions have recently been found. The case study reveals essential features of finding Bellman functions in general and gives the extremal sequences for the problem. Some open questions are posed.

**[B]**Donald L. Burkholder,*Explorations in martingale theory and its applications*, École d’Été de Probabilités de Saint-Flour XIX—1989, Lecture Notes in Math., vol. 1464, Springer, Berlin, 1991, pp. 1–66. MR**1108183**, 10.1007/BFb0085167**[CS]**Mischa Cotlar and Cora Sadosky,*On the Helson-Szegő theorem and a related class of modified Toeplitz kernels*, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 383–407. MR**545279****[N]**F. Nazarov,*A counterexample to a problem of Sarason on boundedness of the product of two Toeplitz operators*, Preprint, 1996, pp. 1-5.**[NTV1]**F. Nazarov, S. Treil, and A. Volberg,*The Bellman functions and two-weight inequalities for Haar multipliers*, J. Amer. Math. Soc.**12**(1999), no. 4, 909–928. MR**1685781**, 10.1090/S0894-0347-99-00310-0**[NTV2]**-,*The T1 theorem for individual Haar multiplier*, Preprint, 2004.**[NV]**F. Nazarov and A. Volberg,*The Bellman function, the two-weight Hilbert transform, and embeddings of the model spaces 𝐾_{𝜃}*, J. Anal. Math.**87**(2002), 385–414. Dedicated to the memory of Thomas H. Wolff. MR**1945290**, 10.1007/BF02868482**[S1]**Eric T. Sawyer,*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), no. 1, 1–11. MR**676801****[S2]**Eric T. Sawyer,*A characterization of two weight norm inequalities for fractional and Poisson integrals*, Trans. Amer. Math. Soc.**308**(1988), no. 2, 533–545. MR**930072**, 10.1090/S0002-9947-1988-0930072-6

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Additional Information

**V. Vasyunin**

Affiliation:
St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia

Email:
vasyunin@pdmi.ras.ru

**A. Volberg**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Email:
volberg@yandex.ru

DOI:
https://doi.org/10.1090/S1061-0022-07-00953-3

Keywords:
Bellman function,
Sawyer test,
two-weight estimate

Received by editor(s):
November 30, 2005

Published electronically:
March 20, 2007

Additional Notes:
The first author was partially supported by RFBR (grant no. 05-01-00925).

The second author was partially supported by NSF (grant DMS 0200713).

Article copyright:
© Copyright 2007
American Mathematical Society