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The Bellman functions and two-weight inequalities for Haar multipliers. (English) Zbl 0951.42007

The main result concerns the two weight problem for Haar multiplier operators. In the one variable case such an operator is the same as a martingale transform \[ T_{\varepsilon,{ a}}: f\mapsto \sum_{I\in {\mathcal D}[0,1]} \varepsilon_I{ a_I} \langle f,h_I\rangle h_I, \] in which \({\mathcal D}[0,1] \) denotes the dyadic intervals in \([0,1]\), \(\{\varepsilon_I\}\) is a sequence of \(\pm 1\)’s indexed by dyadic intervals, \( \{ a_I\}\) is a sequence of nonnegative numbers and the Haar functions are \(h_I= (\chi_{I_l}-\chi_{I_r})/\sqrt{|I|}\) which form an orthonormal basis for \(L^2[0,1]\). Such multipliers are generally thought of as discrete analogues of Calderón-Zygmund singular integral operators such as the Hilbert transform. Establishing boundedness of such operators by means of Carleson type estimates is now considered as major step toward proving boundedness properties of singular integrals. The main result establishes that an operator essentially of the form \( T_{\varepsilon,a}\) is bounded from \( L^2_u\) to \( L^2_v\) if and only if one has Sawyer type estimates \( \|T_{\varepsilon,a} (u\chi_I)\|_{L^2_v}\leq C\|\chi_I\|_{L^2_u}\) and \( \|T_{\varepsilon,a} (v\chi_I)\|_{L^2_u}\leq C\|\chi_I\|_{L^2_v}\) in which the constants do not depend on \(\{\varepsilon_I\}\). Here \(\|f\|_{L^2_v}^2 = \int |f|^2 v\) where \(v\) is a nonnegative weight function. The uniformity of the estimates plays a crucial role. Equally important is the role of weighted Carleson embedding theorems which give a necessary and sufficient Carleson measure type condition for the continuity of the mapping \(f\mapsto \{\langle f\sqrt u\rangle_I\}\) from \(L^2([0,1])\) to a weighted space \(\ell^2_a\) of sequences indexed by dyadic intervals. Here \( \langle \cdot\rangle_I\) denotes the average value over \(I\). Contrary to comments made in the introduction, the authors now appear to have extended their techniques to prove the Sawyer type conditions are necessary and sufficient for boundedness of the Hilbert transform from \( L^2_u\) to \( L^2_v\), at least when the weights satisfy a doubling condition.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42A50 Conjugate functions, conjugate series, singular integrals
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
42B15 Multipliers for harmonic analysis in several variables
60G46 Martingales and classical analysis
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References:

[1] Stephen M. Buckley, Summation conditions on weights, Michigan Math. J. 40 (1993), no. 1, 153 – 170. · Zbl 0794.42011
[2] 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. · Zbl 0771.60033
[3] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241 – 250. · Zbl 0291.44007
[4] R. R. Coifman, Peter W. Jones, and Stephen Semmes, Two elementary proofs of the \?² boundedness of Cauchy integrals on Lipschitz curves, J. Amer. Math. Soc. 2 (1989), no. 3, 553 – 564. · Zbl 0713.42010
[5] 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. · Zbl 0448.42008
[6] M. Cotlar and C. Sadosky, On some \?^{\?} 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.
[7] 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. · Zbl 0575.42025
[8] Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129 – 206. · Zbl 0526.35080
[9] 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. · Zbl 0770.35014
[10] John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. · Zbl 0469.30024
[11] N.J. Kalton, J.E. Verbitsky, Nonlinear equations and weighted norm inequalities, Trans. Amer. Math. Soc., to appear. CMP 98:02 · Zbl 0948.35044
[12] F. Nazarov, A counterexample to a problem of Sarason on boundedness of the product of two Toeplitz operators. Preprint, 1996, 1-5.
[13] F. L. Nazarov and S. R. Treĭl\(^{\prime}\), 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. · Zbl 0873.42011
[14] 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 · Zbl 0858.44004
[15] F. Nazarov, S. Treil, and A. Volberg, Cauchy integral and Calderón-Zygmund operators on nonhomogeneous spaces, Internat. Math. Res. Notices 15 (1997), 703 – 726. · Zbl 0889.42013
[16] F.Nazarov, S.Treil, A.Volberg, The Bellman functions and two weight inequalities for Haar multipliers, MSRI Preprint 1997-103, p. 1-31.
[17] C. J. Neugebauer, Inserting \?_{\?}-weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 644 – 648. · Zbl 0521.42019
[18] 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. · Zbl 1128.47309
[19] Eric T. Sawyer, Norm inequalities relating singular integrals and the maximal function, Studia Math. 75 (1983), no. 3, 253 – 263. · Zbl 0528.44002
[20] Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1 – 11. · Zbl 0508.42023
[21] 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. · Zbl 0665.42023
[22] 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. · Zbl 0821.42001
[23] 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. · Zbl 0783.42011
[24] X. Tolsa, Boundedness of the Cauchy integral operator. Preprint, 1997.
[25] 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 · Zbl 0876.42027
[26] 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. · Zbl 0852.42006
[27] 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. CMP 98:11 · Zbl 0896.42009
[28] J.E. Verbitsky, R.L. Wheeden, Weighted norm inequalities for integral operators. Preprint, 1996. 1-25.
[29] A. Volberg, Matrix \?_{\?} weights via \?-functions, J. Amer. Math. Soc. 10 (1997), no. 2, 445 – 466. · Zbl 0877.42003
[30] Dechao Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators, J. Funct. Anal. 138 (1996), no. 2, 477 – 501. · Zbl 0865.47019
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