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Relations Between Product and Flag Hardy Spaces

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Abstract

In Nagel et al. (J Funct Anal 181:29–118, 2001), Nagel–Ricci–Stein established the relationships between product kernels and flag kernels on the Euclidean space, that is, product kernels are finite sums of flag kernels. The main purpose of this paper is to characterize the product Hardy space as the intersection of flag Hardy spaces, and characterize the product Carleson measure space as the sum of flag spaces.

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References

  1. Bergh, J., Löfström, J.: Interpolation Spaces. Springer, New York (1976)

    Book  MATH  Google Scholar 

  2. Calderón, A.: Intermediate spaces and interpolation, the complex method. Studia Math. 24, 113–190 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, S.-Y.A.: Carleson measure on the bi-disc. Ann. Math. 109, 613–620 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  4. Chang, S.-Y.A., Fefferman, R.: A continuous version of duality of \(H^1\) with BMO on the bidisc. Ann. Math. 112, 179–201 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Chang, S.-Y.A., Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Am. J. Math. 104, 455–468 (1982)

    Article  MATH  Google Scholar 

  6. Chang, S.-Y.A., Fefferman, R.: Some recent developments in Fourier analysis and Hp-theory on product domains. Bull. Am. Math. Soc. (N.S.) 12, 1–43 (1985)

    Article  MATH  Google Scholar 

  7. David, G., Journé, J.L., Semmes, S.: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoam. 1, 1–56 (1985)

    Article  MATH  Google Scholar 

  8. Ding, Y., Lu, G.Z., Ma, B.L.: Multi-parameter Triebel-Lizorkin and Besov spaces associated with flag singular integrals. Acta Math. Sin. (Engl. Ser.) 26, 603–620 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duong, X., Li, J., Ou, Y., Pipher, J., Wick, B.D.: Commutators of multiparameter flag singular integrals and applications. Anal. PDE 12, 1325–1355 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fefferman, C.: Characterizations of bounded mean oscillation. Bull. Am. Math. Soc. 77, 587–588 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fefferman, R.: Calderón-Zygmund theory for product domains: \(H^p\) spaces. Proc. Nat. Acad. Sci. U.S.A. 83, 840–843 (1986)

    MathSciNet  MATH  Google Scholar 

  12. Fefferman, R.: Harmonic analysis on product spaces. Ann. Math. 126, 109–130 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fefferman, R.: \(A^p\) weights and singular integrals. Am. J. Math. 110, 975–987 (1988)

    Article  MATH  Google Scholar 

  14. Fefferman, C., Stein, E.M.: \(H^p\) spaces of several variables. Acta Math. 129, 137–193 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  15. Fefferman, R., Stein, E.M.: Singular integrals on product spaces. Adv. Math. 45, 117–143 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ferguson, S.H., Lacey, M.: A characterization of product BMO by commutators. Acta Math. 189, 143–160 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution. J. Funct. Anal. 93, 34–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series 79. A.M.S, Providence, RI (1991)

  19. Garcia-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam (1985)

    MATH  Google Scholar 

  20. Głowacki, P.: Composition and \(L^2\)-boundedness of flag kernels. Colloq. Math. 118, 581–585 (2010)

    MathSciNet  MATH  Google Scholar 

  21. Głowacki, P.: Correction to “Composition and \(L^2\)-boundedness of flag kernels”. Colloq. Math. 120, 331 (2010)

    MathSciNet  MATH  Google Scholar 

  22. Głowacki, P.: \(L^p\)-boundedness of flag kernels on homogeneous groups via symbolic calculus. J. Lie Theory 23, 953–977 (2013)

    MathSciNet  MATH  Google Scholar 

  23. Gundy, R., Stein, E.M.: \(H^p\) spaces on the polydisc. Proc. Natl. Acad. Sci. 76, 1026–1029 (1979)

    Article  MATH  Google Scholar 

  24. Han, Y., Lu, G.: Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with flag singular integrals. arXiv:0801.1701

  25. Han, Y., Sawyer, E.: Littlewood-Paley-Stein theorem on space of homogeneous type and classical function spaces. Mem. Am. Math. Soc. 110, 1–126 (1994)

    Google Scholar 

  26. Han, Y., Lee, M.-Y., Lin, C.-C., Lin, Y.-C.: Calderón-Zygmund operators on product Hardy spaces. J. Funct. Anal. 258, 2834–2861 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Han, Y., Lin, C.-C., Wu, X.: Boundedness of singular integrals with flag kernels on weighted flag Hardy spaces. Pacific J. Math., to appear

  28. Han, Y., Lu, G., Sawyer, E.: Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group. Anal. PDE 7, 1465–1534 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. Journé, J.L.: Calderón-Zygmund operators on product space. Rev. Mat. Iberoam. 1, 55–91 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  30. Journé, J.L.: Two problems of Calderón-Zygmund theory on product-spaces. Ann. Inst. Fourier (Grenoble) 38, 111–132 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kalton, N.: Basic sequences in \(F\)-spaces and their applications, Proc. Edinburgh Math. Soc. (2) 19 151–167 (1974/1975)

  32. Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, I. Invent. Math. 119, 119–233 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, II. Math. Z. 221, 267–291 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  34. Nagel, A., Stein, E.M.: The \({\bar{\partial }}_b\)-complex on decoupled boundaries in \({\mathbb{C}}^ n\). Ann. Math. 164, 649–713 (2006)

    MathSciNet  MATH  Google Scholar 

  35. Nagel, A., Ricci, F., Stein, E.M.: Singular integrals with flag kernels and analysis on quadratic CR manifolds. J. Funct. Anal. 181, 29–118 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  36. Nagel, A., Ricci, F., Stein, E.M., Wainger, S.: Singular integrals with flag kernels on homogeneous groups: I. Rev. Mat. Iberoam. 28, 631–722 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  37. Nagel, A., Ricci, F., Stein, E.M., Wainger, S.: Algebras of singular integral operators with kernels controlled by multiple norms. Mem. Am. Math. Soc. 256(1230) (2018). https://doi.org/10.1090/memo/1230

  38. Phong, D.H., Stein, E.M.: Some further classes of pseudo-differential and singular integral operators arising in boundary valve problems I, composition of operators. Am. J. Math. 104, 141–172 (1982)

    Article  MATH  Google Scholar 

  39. Pipher, J.: Journé’s covering lemma and its extension to higher dimensions. Duke Math. J. 53, 683–690 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  40. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  41. Stein, E.M.: Singular integrals: the roles of Calderón and Zygmund. Notices of the AMS 45, 1130–1140 (1998)

    MATH  Google Scholar 

  42. Wu, X.: An atomic decomposition characterization of flag Hardy spaces \(H^p_F({\mathbb{R}}^{n}\times {\mathbb{R}}^m)\) with applications. J. Geom. Anal. 24, 613–626 (2014)

    MathSciNet  Google Scholar 

  43. Wu, X.: Weighted norm inequalities for flag singular integrals on homogeneous groups. Taiwanese J. Math. 18, 357–369 (2014)

    MathSciNet  MATH  Google Scholar 

  44. Wu, X., Liu, Z.: Characterizations of multiparameter Besov and Triebel-Lizorkin spaces associated with flag singular integrals. J. Funct. Spaces Appl. 1, 1 (2012). https://doi.org/10.1155/2012/275791

    Article  MathSciNet  MATH  Google Scholar 

  45. Yang, D.: Besov and Triebel-Lizorkin spaces related to singular integrals with flag kernels. Rev. Mat. Complut. 22, 253–302 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Georgetown University, Washington, DC, 20057, USA

    Der-Chen Chang

  2. Graduate Institute of Business Administration, College of Management, Fu Jen Catholic University, New Taipei City, 242, Taiwan, ROC

    Der-Chen Chang

  3. Department of Mathematics and Statistics, Auburn University, Auburn, AL, 36849, USA

    Yongsheng Han

  4. Department of Mathematics, China University of Mining & Technology, Beijing, 100083, China

    Xinfeng Wu

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  1. Der-Chen Chang
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  2. Yongsheng Han
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  3. Xinfeng Wu
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Correspondence to Xinfeng Wu.

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Chang, DC., Han, Y. & Wu, X. Relations Between Product and Flag Hardy Spaces. J Geom Anal 31, 6601–6623 (2021). https://doi.org/10.1007/s12220-019-00284-2

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  • DOI: https://doi.org/10.1007/s12220-019-00284-2

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