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

HTML articles powered by AMS MathViewer

- by F. Nazarov, S. Treil and A. Volberg PDF
- J. Amer. Math. Soc.
**12**(1999), 909-928 Request permission

## 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

- Stephen M. Buckley,
*Summation conditions on weights*, Michigan Math. J.**40**(1993), no. 1, 153–170. MR**1214060**, DOI 10.1307/mmj/1029004679 - 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**, DOI 10.1007/BFb0085167 - R. R. Coifman and C. Fefferman,
*Weighted norm inequalities for maximal functions and singular integrals*, Studia Math.**51**(1974), 241–250. MR**358205**, DOI 10.4064/sm-51-3-241-250 - R. R. Coifman, Peter W. Jones, and Stephen Semmes,
*Two elementary proofs of the $L^2$ boundedness of Cauchy integrals on Lipschitz curves*, J. Amer. Math. Soc.**2**(1989), no. 3, 553–564. MR**986825**, DOI 10.1090/S0894-0347-1989-0986825-6 - 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** - M. Cotlar and C. Sadosky,
*On some $L^{p}$ versions of the Helson-Szegő theorem*, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 306–317. MR**730075** - S.-Y. A. Chang, J. M. Wilson, and T. H. Wolff,
*Some weighted norm inequalities concerning the Schrödinger operators*, Comment. Math. Helv.**60**(1985), no. 2, 217–246. MR**800004**, DOI 10.1007/BF02567411 - Charles L. Fefferman,
*The uncertainty principle*, Bull. Amer. Math. Soc. (N.S.)**9**(1983), no. 2, 129–206. MR**707957**, DOI 10.1090/S0273-0979-1983-15154-6 - R. A. Fefferman, C. E. Kenig, and J. Pipher,
*The theory of weights and the Dirichlet problem for elliptic equations*, Ann. of Math. (2)**134**(1991), no. 1, 65–124. MR**1114608**, DOI 10.2307/2944333 - John B. Garnett,
*Bounded analytic functions*, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR**628971** - N.J. Kalton, J.E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc., to appear.
- F. Nazarov, A counterexample to a problem of Sarason on boundedness of the product of two Toeplitz operators. Preprint, 1996, 1-5.
- F. L. Nazarov and S. R. Treĭl′,
*The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis*, Algebra i Analiz**8**(1996), no. 5, 32–162 (Russian, with Russian summary); English transl., St. Petersburg Math. J.**8**(1997), no. 5, 721–824. MR**1428988** - F.Nazarov, S.Treil, The weighted norm inequalities for Hilbert transform are now trivial, C.R. Acad. Sci. Paris, Série J,
**323**, (1996), 717-722. - F. Nazarov, S. Treil, and A. Volberg,
*Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces*, Internat. Math. Res. Notices**15**(1997), 703–726. MR**1470373**, DOI 10.1155/S1073792897000469 - F.Nazarov, S.Treil, A.Volberg, The Bellman functions and two weight inequalities for Haar multipliers, MSRI Preprint 1997-103, p. 1-31.
- C. J. Neugebauer,
*Inserting $A_{p}$-weights*, Proc. Amer. Math. Soc.**87**(1983), no. 4, 644–648. MR**687633**, DOI 10.1090/S0002-9939-1983-0687633-2 - Cora Sadosky,
*Liftings of kernels shift-invariant in scattering systems*, Holomorphic spaces (Berkeley, CA, 1995) Math. Sci. Res. Inst. Publ., vol. 33, Cambridge Univ. Press, Cambridge, 1998, pp. 303–336. MR**1630653** - Eric T. Sawyer,
*Norm inequalities relating singular integrals and the maximal function*, Studia Math.**75**(1983), no. 3, 253–263. MR**722250**, DOI 10.4064/sm-75-3-253-263 - Eric T. Sawyer,
*A characterization of a two-weight norm inequality for maximal operators*, Studia Math.**75**(1982), no. 1, 1–11. MR**676801**, DOI 10.4064/sm-75-1-1-11 - 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**, DOI 10.1090/S0002-9947-1988-0930072-6 - Elias M. Stein,
*Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals*, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR**1232192** - E. Sawyer and R. L. Wheeden,
*Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces*, Amer. J. Math.**114**(1992), no. 4, 813–874. MR**1175693**, DOI 10.2307/2374799 - X. Tolsa, Boundedness of the Cauchy integral operator. Preprint, 1997.
- S. Treil and A. Volberg, Wavelets and the angle between past and future, J. Funct. Anal.
**143**(1997), no. 2, 269–308. - S. R. Treil and A. L. Volberg,
*Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator*, Algebra i Analiz**7**(1995), no. 6, 205–226; English transl., St. Petersburg Math. J.**7**(1996), no. 6, 1017–1032. MR**1381983** - S.R. Treil, A.L. Volberg, D. Zheng, Hilbert transform, Toeplitz operators and Hankel operators, and invariant $A_{\infty }$ weights. Revista Mat. Iberoamericana,
**13**(1997), No. 2, 319-360. - J.E. Verbitsky, R.L. Wheeden, Weighted norm inequalities for integral operators. Preprint, 1996. 1-25.
- A. Volberg,
*Matrix $A_p$ weights via $S$-functions*, J. Amer. Math. Soc.**10**(1997), no. 2, 445–466. MR**1423034**, DOI 10.1090/S0894-0347-97-00233-6 - Dechao Zheng,
*The distribution function inequality and products of Toeplitz operators and Hankel operators*, J. Funct. Anal.**138**(1996), no. 2, 477–501. MR**1395967**, DOI 10.1006/jfan.1996.0073

## Additional 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