The Bellman functions and two-weight inequalities for Haar multipliers

Authors:
F. Nazarov, S. Treil and A. Volberg

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

Published electronically:
June 24, 1999

MathSciNet review:
1685781

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

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Additional Information

**F. Nazarov**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
fedja@math.msu.edu

**S. Treil**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
treil@math.msu.edu

**A. Volberg**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
volberg@math.msu.edu

DOI:
https://doi.org/10.1090/S0894-0347-99-00310-0

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.

Article copyright:
© Copyright 1999
American Mathematical Society