×

Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem. (English) Zbl 0961.44006

From the introduction: This paper deals with the weighted norm inequalities for the Hilbert transform with matrix-valued weights. The main problem can be formulated as follows. Let \(W\) be a \(d\times d\) matrix weight, i.e. an \(L^1\) function whose values are selfadjoint nonnegative \(d\times d\) matrices. We suppose that the weight \(W\) is defined on the unit circle \(T=\{z \in\mathbb{C}: |z|= 1\}\).
Let \(L^2=L^2 (\mathbb{C}^d)\) be the space of square summable functions on \(T\) with values in \(\mathbb{C}^d\), let \(H^2=H^2 (\mathbb{C}^d)\) be the corresponding Hardy space of analytic functions, and let \(P_+\) be the orthogonal projection in \(L^2\) onto \(H^2\). Let \(T\) denote the Hilbert transform, \(T=-iP_+ +i(I-P_+)\).
The question we are interested in is under what conditions on \(W\) the following weighted norm inequality for \(T\) holds (say for all \(f\in L^2\cap L^\infty)\), \[ \int_T\bigl(W(\xi) Tf(\xi), Tf(t)\bigr)dt\leq C \int_T\bigl(W(\xi) f(\xi), f(\xi)\bigr) dm(\xi), \] where \(m\) is the normalized \((m(T)=1)\) Lebesgue measure on \(T\).

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] [ACS]Axler, S., Chang, S.-Y. A. andSarason, D., Products of Toeplitz operators,Integral Equations Operator Theory 1 (1978), 285–309. · Zbl 0396.47017
[2] [B1]Bloom, S., A commutator theorem and weighted BMO,Trans. Amer. Math. Soc. 292 (1985), 103–122. · Zbl 0578.42012
[3] [B2]Bloom, S., Applications of commutator theory to weighted BMO and matrix analogs ofA 2,Illinois J. Math. 33 (1989), 464–487. · Zbl 0661.42013
[4] [D]Daubechies, I.,Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math.61, Soc. Ind. Appl. Math., Philadelphia, Pa., 1992.
[5] [Do]Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space,Proc. Amer. Math. Soc. 17 (1966), 413–415. · Zbl 0146.12503
[6] [GCRF]García-Cuerva, J. andRubio de Francia, J. L.,Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam-New York, 1985.
[7] [G]Garnett, J. B.,Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981. · Zbl 0469.30024
[8] [HS]Helson, H. andSarason, D., Past and future,Math. Scand.,21 (1967), 5–16.
[9] [HMW]Hunt, R. A., Muckenhoupt, B. andWheeden, R. L., Weighted norm inequalities for the conjugate function and the Hilbert transform,Trans. Amer. Math. Soc. 176 (1973), 227–251.
[10] [KS]Kolmogorov, A. andSeliverstov, G., Sur la convergence de series de Fourier,C. R. Acad. Sci. Paris 178 (1925), 303–305.
[11] [NT]Nazarov, F. andTreil, S., The hunt for a Bellman function: applications to estimates of singular integral operators and to other classical problems in harmonic analysis,Algebra i Analiz 8:5 (1996), 32–162.
[12] [N]Nikolskii, N. K.,Treatise on the Shift Operator, Springer-Verlag, Berlin-New York, 1986.
[13] [R]Rozanov, Yu. A.,Stationary Stochastic Processes, Holden-Day, San Francisco, Calif., 1967. · Zbl 0152.16302
[14] [S1]Sarason, D., Exposed points inH 1. II, inTopics in Operator Theory: Ernst D. Hellinger Memorial Volume (de Branges, L., Gohberg, I. and Rovnyak, J., eds.), Oper. Theory Adv. Appl.48, pp. 333–347, Birkhäuser, Basel, 1990.
[15] [S2]Sarason, D., Products of Toeplitz operators, inLinear and Complex Analysis Problem Book 3, Part I (Havin, V. P. and Nikolski, N. K., eds.). Lecture Notes in Math.1573, pp. 318–319, Springer-Verlag, Berlin-Heidelberg, 1994.
[16] [Si]Simonenko, I. B., Riemann’s boundary value problem forn pairs of functions with measurable coefficients and its applications to the study of singular integrals inL p spaces with weights,Dokl Akad. Nauk SSSR 141 (1961), 36–39 (Russian). English transl.:Soviet Math. Dokl. 2 (1961), 1391–1394.
[17] [St]Stein, E.,Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press, Princeton, N. J., 1993. · Zbl 0821.42001
[18] [T]Treil, S., Geometric methods in spectral theory of vector valued functions: Some recent results, inToeplitz Operators and Spectral Function Theory (Nikolskii, N. K., ed.), Oper. Theory Adv. Appl.42, pp. 209–280, Birkhäuser, Basel, 1989.
[19] [TV1]Treil, S. andVolberg, A., Wavelets and the angle between past and future,J. Funct. Anal. 143 (1997), 269–308. · Zbl 0876.42027
[20] [TV2]Treil, S. andVolberg, A., A simple proof of Hunt-Muckenhoupt-Wheeden theorem,Preprint, 1995.
[21] [TV3]Treil, S. andVolberg, A., Completely regular multivariate processes and matrix weighted estimates,Preprint, 1996.
[22] [TVZ]Treil, S. Volberg, A. andZheng, D., Hilbert transform, Toeplitz operators and Hankel operators, and invariant Aweights, to appear inRev. Mat. Iberoamericana.
[23] [V]Volberg, A., MatrixA p weights viaS-functions, to appear inJ. Amer. Math. Soc.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.