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.

**[B]**St. Buckley, Summation conditions on weights, Mich. Math. J.,**40**(1993), 153-170. MR**94d:42021****[Bu]**D.L Burkholder, Explorations in martingale theory and its applications. Ecole d'Eté de Probabilité de Saint-Flour XIX-1989, 1-66, Lecture Notes in Mathematics,**1464**, Springer, Berlin, 1991. MR**92m:60037****[CF]**R.R. Coifman, Ch. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math.,**51**(1974), 241-250. MR**50:10670****[CJS]**R.R. Coifman, P.W. Jones, and St. Semmes, Two elementary proofs of the boundedness of Cauchy integrals on Lipschitz curves, J. of Amer. Math. Soc.,**2**(1989), No. 3, 553-564. MR**90k:42017****[CS1]**M. Cotlar, C. Sadosky, On the Helson-Szegö theorem and a related class of modified Toeplitz kernels, in Harmonic Analysis in Euclidean spaces, ed. by G.Weiss and S. Wainger, Proc. Symp. Pure Math.**35**, Amer. Math. Soc., Providence, R.I., 1979, 383-407. MR**81j:42022****[CS2]**M. Cotlar, C. Sadosky, On some version of the Helson-Szegö theorem, Conference on Harmonic Analysis in honor of Antony Zygmund (Chicago, 1981),**vol.1**, ed. by W. Beckner et al., Wadsworth Math. Ser,. Wadsworth, Belmont, CA, 1983, 306-317. MR**85i:42015****[ChWW]**A. Chang, J.M. Wilson, Th. Wolff, Some weighted norm inequalities concerning the Schrödinger operators. Comment. Math. Helvetici,**60**(1985), 217-246. MR**87d:42027****[F]**C. Fefferman, The uncertainty principle. Bull. of Amer. Math. Soc.,**9**(1983), No. 2, 127-206. MR**85f:35001****[FKP]**R.A. Fefferman, C.E. Kenig, J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math.**134**(1991), 65-124. MR**93h:31010****[G]**John B. Garnett, Bounded analytic functions. Pure and Applied Mathematics, 96. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. xvi+467 MR**83g:30037****[KV]**N.J. Kalton, J.E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc., to appear. CMP**98:02****[N]**F. Nazarov, A counterexample to a problem of Sarason on boundedness of the product of two Toeplitz operators. Preprint, 1996, 1-5.**[NT]**F. L. Nazarov and S. R. Treil, 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. MR**99d:42026****[NT1]**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. CMP**97:03****[NTV]**F.Nazarov, S.Treil, A.Volberg, Cauchy integral and Calderón-Zygmund operators on non-homogeneous spaces. IMRN (Int. Math. Res. Notes.),**1997**, No. 15, 703-726. MR**99e:42028****[NTV1]**F.Nazarov, S.Treil, A.Volberg, The Bellman functions and two weight inequalities for Haar multipliers, MSRI Preprint 1997-103, p. 1-31.**[Ne]**C.J. Neugebauer, Inserting -weights, Proc. of the Amer. Math. Soc.**87**(1983), 644-648. MR**84d:42026****[S]**C. Sadosky, Lifting of kernels shift-invariant in scattering systems, Holomorphic spaces, MSRI publications,**32**, 1997. MR**99e:47034****[Sa]**E.T. Sawyer, Norm inequalities relating singular integrals and the maximal functions, Studia Math.**75**(1983), No. 3, 253-263. MR**85c:42018****[S1]**E.T. Sawyer, A characterization of a two weight norm inequality for maximal operators, Studia Math.**75**(1982), 1-11. MR**84i:42032****[S2]**E.T. Sawyer, A characterization of two weight norm inequality for fractional and Poisson integrals. Trans. Amer. Math. Soc.**308**(1988), 533-545. MR**89d:26009****[St]**E.M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Math. Series, 43, Monographs in Harmonic Analysis, Princeton Univ. Press, Princeton, NJ, 1993. MR**95c:42002****[SW]**E.T. Sawyer, R.L.Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math.**114**(1992), 813-874. MR**94i:42024****[T]**X. Tolsa, Boundedness of the Cauchy integral operator. Preprint, 1997.**[TV1]**S. Treil and A. Volberg, Wavelets and the angle between past and future, J. Funct. Anal.**143**(1997), no. 2, 269-308. CMP**97:06****[TV2]**S.R. Treil, A.L. Volberg, Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator. St. Petersburg Math. J.**7**(1996), 207-226. MR**97c:42017****[TVZ]**S.R. Treil, A.L. Volberg, D. Zheng, Hilbert transform, Toeplitz operators and Hankel operators, and invariant weights. Revista Mat. Iberoamericana,**13**(1997), No. 2, 319-360. CMP**98:11****[VW]**J.E. Verbitsky, R.L. Wheeden, Weighted norm inequalities for integral operators. Preprint, 1996. 1-25.**[V]**A. Volberg, Matrix weights via -functions. J. Amer. Math. Soc.**10**(1997), 445-466. MR**98a:42013****[Zh]**Dechao Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators, J. Funct. Anal.**138**(1996), no. 2, 477-501. MR**97e:47040**

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
42B20,
42A50,
47B35

Retrieve articles in all journals with MSC (1991): 42B20, 42A50, 47B35

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