St. Petersburg Mathematical Society

Author(s): V. Vasyunin; 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
Posted: March 20, 2007
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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]: D. 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 (92m:60037)
[CS]: M. Cotlar and C. Sadosky, On the Helson-Szegö theorem and a related class of modified Toeplitz kernels, Harmonic Analysis in Euclidean Spaces (G. Weiss and S. Wainger, eds.) (Proc. Sympos. Pure Math., Williams Coll., Williamstown, MA, 1978), Part 1, Proc. Sympos. Pure Math., vol. 35, Amer. Math. Soc., Providence, RI, 1979, pp. 383-407. MR 545279 (81j:42022)
[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), 909-928. MR 1685781 (2000k:42009)
[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 $K_{\theta}$ . Dedicated to the memory of Thomas H. Wolff, J. Anal. Math. 87 (2002), 385-414. MR 1945290 (2003j:30081)
[S1]: E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), 1-11. MR 676801 (84i:42032)
[S2]: -, A characterization of two weight norm inequalities for fractional and Poisson integrals, Trans. Amer. Math. Soc. 308 (1988), 533-545. MR 930072 (89d:26009)

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 42B20, 42A50, 47B35

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: 10.1090/S1061-0022-07-00953-3
PII: S 1061-0022(07)00953-3
Keywords: Bellman function, Sawyer test, two-weight estimate
Received by editor(s): 30/NOV/2005
Posted: 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).
Copyright of article: Copyright 2007, American Mathematical Society

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ISSN: 1547-7371(e) ISSN: 1061-0022(p)

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