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Boundedness of paraproduct operators on RD-spaces

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Abstract

Let (X, d,µ) be an RD-space with “dimension” n, namely, a space of homogeneous type in the sense of Coifman and Weiss satisfying a certain reverse doubling condition. Using the Calderón reproducing formula, the authors hereby establish boundedness for paraproduct operators from the product of Hardy spaces H p(X) × H q(X) to the Hardy space H r(X), where p, q, r ∈ (n/(n + 1),∞) satisfy 1/p + 1/q = 1/r. Certain endpoint estimates are also obtained. In view of the lack of the Fourier transform in this setting, the proofs are based on the derivation of appropriate kernel estimates.

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References

  1. Alexopoulos G. Spectral multipliers on Lie groups of polynomial growth. Proc Amer Math Soc, 1994, 120: 973–979

    MATH  MathSciNet  Google Scholar 

  2. Auscher P, Hofmann S, Muscalu C, et al. Carleson measures, trees, extrapolation, and T(b) theorems. Publ Mat, 2002, 46: 257–325

    MATH  MathSciNet  Google Scholar 

  3. Christ M. AT(b) theorem with remarks on analytic capacity and the Cauchy integral. Colloq Math, 1990, 60/61: 601–628

    MathSciNet  Google Scholar 

  4. Coifman R R, Meyer Y. Commutateurs d’intégrales singulières et opérateurs multilinéaires. Ann Inst Fourier (Grenoble), 1978, 28: 177–202

    MATH  MathSciNet  Google Scholar 

  5. Coifman R R, Meyer Y. Au delà des opérateurs pseudo-différentiels. Astérisque, 1979, 57: 1–185

    Google Scholar 

  6. Coifman R R, Weiss G. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math., Vol. 242. Berlin: Springer, 1971

    MATH  Google Scholar 

  7. Coifman R R, Weiss G. Extensions of Hardy spaces and their use in analysis. Bull Amer Math Soc, 1977, 83: 569–645

    Article  MATH  MathSciNet  Google Scholar 

  8. David G, Journé J L. A boundedness criterion for generalized Calderón-Zygmund operators. Ann of Math (2), 1984, 120: 371–397

    Article  MathSciNet  Google Scholar 

  9. Duong X T, Yan L X. Hardy spaces of spaces of homogeneous type. Proc Amer Math Soc, 2003, 131: 3181–3189

    Article  MATH  MathSciNet  Google Scholar 

  10. Grafakos L, Kalton N. The Marcinkiewicz multilinear condition for bilinear operators. Studia Math, 2001, 146: 115–156

    Article  MATH  MathSciNet  Google Scholar 

  11. Grafakos L, Liu L, Yang D. Maximal function characterizations of Hardy spaces on RD-spaces and their applications. Sci China Ser A, 2008, 51: 2253–2284

    Article  MATH  MathSciNet  Google Scholar 

  12. Grafakos L, Liu L, Yang D. Vector-valued singular integrals and maximal functions on spaces of homogeneous type. Math Scand, 2009, 104: 296–310

    MATH  MathSciNet  Google Scholar 

  13. Grafakos L, Torres R H. Discrete decompositions for bilinear operators and almost diagonal conditions. Trans Amer Math Soc, 2001, 354: 1153–1176

    Article  MathSciNet  Google Scholar 

  14. Grafakos L, Torres R H. Multilinear Calderón-Zygmund theory. Adv Math, 2002, 165: 124–164

    Article  MATH  MathSciNet  Google Scholar 

  15. Han Y. Triebel-Lizorkin spaces on spaces of homogeneous type. Studia Math, 1994, 108: 247–273

    MATH  MathSciNet  Google Scholar 

  16. Han Y, Müller D, Yang D. Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type. Math Nachr, 2006, 279: 1505–1537

    Article  MATH  MathSciNet  Google Scholar 

  17. Han Y, Müller D, Yang D. A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr Appl Anal, 2008, Art. ID 893409: 250 pp

  18. Macías R A, Segovia C. A decomposition into atoms of distributions on spaces of homogeneous type. Adv Math, 1979, 33: 271–309

    Article  MATH  Google Scholar 

  19. Muscalu C, Pipher J, Tao T, et al. A short proof of the Coifman-Meyer paraproduct theorem. Preprint, available at http://www.math.brown.edujpipher/trilogy1.pdf

  20. Nagel A, Stein E M. Differentiable control metrics and scaled bump functions. J Differential Geom, 2001, 57: 465–492

    MATH  MathSciNet  Google Scholar 

  21. Nagel A, Stein E M. On the product theory of singular integrals. Rev Mat Iberoamericana, 2004, 20: 531–561

    MathSciNet  Google Scholar 

  22. Nagel A, Stein E M, Wainger S. Balls and metrics defined by vector fields I. Basic properties. Acta Math, 1985, 155: 103–147

    Article  MATH  MathSciNet  Google Scholar 

  23. Stein E M. Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton University Press, 1993

    MATH  Google Scholar 

  24. Stein E M. Some geometrical concepts arising in harmonic analysis. Geom Funct Anal, 2000, 10: 434–453

    Google Scholar 

  25. Varopoulos N Th. Analysis on Lie groups. J Funct Anal, 1988, 76: 346–410

    Article  MATH  MathSciNet  Google Scholar 

  26. Varopoulos N Th, Saloff-Coste L, Coulhon T. Analysis and Geometry on Groups. Cambridge: Cambridge University Press, 1992

    Google Scholar 

  27. Yang D, Zhou Y. New properties of Besov and Triebel-Lizorkin spaces on RD-spaces. Manuscript Math, to appear

  28. Yang D, Zhou Y. Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms. J Math Anal Appl, 2008, 339: 622–635

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Loukas Grafakos

  2. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, China

    LiGuang Liu & DaChun Yang

Authors
  1. Loukas Grafakos
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  2. LiGuang Liu
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  3. DaChun Yang
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Correspondence to DaChun Yang.

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Grafakos, L., Liu, L. & Yang, D. Boundedness of paraproduct operators on RD-spaces. Sci. China Math. 53, 2097–2114 (2010). https://doi.org/10.1007/s11425-010-4042-3

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