## The Bellman functions and two-weight inequalities for Haar multipliers

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- by F. Nazarov, S. Treil and A. Volberg
- J. Amer. Math. Soc.
**12**(1999), 909-928 - DOI: https://doi.org/10.1090/S0894-0347-99-00310-0
- Published electronically: June 24, 1999
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## Abstract:

We give necessary and sufficient conditions for two-weight norm inequalities for Haar multiplier operators and for square functions. The conditions are of the type used by Eric Sawyer in characterizing the boundedness of the wide class of operators with positive kernel. The difference is that our operator is essentially singular. We also show how to separate two Sawyer’s conditions (even for positive kernel operators) by finding which condition is responsible for which estimate.## References

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## Bibliographic Information

**F. Nazarov**- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
- MR Author ID: 233855
- Email: fedja@math.msu.edu
**S. Treil**- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
- MR Author ID: 232797
- Email: treil@math.msu.edu
**A. Volberg**- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
- Email: volberg@math.msu.edu
- Received by editor(s): December 31, 1997
- Published electronically: June 24, 1999
- Additional Notes: This work was partially supported by NSF grant DMS 9622936, the joint Israeli-USA grant BSF 00030, and MSRI programs of the Fall 1995 and the Fall 1997.
- © Copyright 1999 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**12**(1999), 909-928 - MSC (1991): Primary 42B20, 42A50, 47B35
- DOI: https://doi.org/10.1090/S0894-0347-99-00310-0
- MathSciNet review: 1685781