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Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type


Authors: Yongsheng Han, Ji Li and Guozhen Lu
Journal: Trans. Amer. Math. Soc. 365 (2013), 319-360
MSC (2010): Primary 42B35; Secondary 32T25, 32W30
DOI: https://doi.org/10.1090/S0002-9947-2012-05638-8
Published electronically: July 24, 2012
MathSciNet review: 2984061
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Abstract: This paper is inspired by the work of Nagel and Stein in which the $ L^p$ $ (1<p<\infty )$ theory has been developed in the setting of the product Carnot-Carathéodory spaces $ \widetilde {M}=M_1\times \cdots \times M_n$ formed by vector fields satisfying Hörmander's finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product $ BMO$ space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.


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  • 1. A. Boggess, R. Dwilewicz and A. Nagel, The hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type, Trans. Amer. Math. Soc. 323 (1991), 209-232. MR 1079050 (92a:32017)
  • 2. A. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. MR 0167830 (29:5097)
  • 3. A. Carbery and A. Seeger, $ H^p$ and $ L^p$ variants of multiparameter Calderón-Zygmund theory, Transaction of Amer. Math. Soc. 324(2), 719-747. MR 1072104 (93b:42035)
  • 4. M. Christ, A $ T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), 601-628. MR 1096400 (92k:42020)
  • 5. M. Christ, Regularity properties of the $ \overline {\partial }_b$ equation on weakly pseudoconvex CR manifolds of dimension $ 3$, J. Amer. Math. Soc. 1 (1988), 587-646. MR 928903 (89e:32027)
  • 6. M. Christ, On the $ \overline {\partial }_b$ equation for three-dimensional CR manifolds, Proc. Sympos. Pure Math., Amer. Math. Soc. 52 (3), (1991), 63-82. MR 1128584 (92g:32035)
  • 7. S. Y. Chang and R. Fefferman, A continuous version of the duality of $ H^1$ and $ BMO$ on the bidisc, Ann. of Math. 112 (1980), 179-201. MR 584078 (82a:32009)
  • 8. S. A. Chang and R. Fefferman, The Calderón-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), 455-468. MR 658542 (84a:42028)
  • 9. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971. MR 0499948 (58:17690)
  • 10. D.G. Deng, Y.S. Han, Harmonic Analysis on Spaces of Homogeneous Type, Lecture Notes in Mathematics, 1966, Springer-Verlag, Berlin, 2009. MR 2467074 (2010i:43005).
  • 11. G. David, J.L. Journé and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoam. 1 (1985) 1-56. MR 850408 (88f:47024)
  • 12. C. Fefferman and J.J. Kohn, Hölder estimates on domains of complex dimension two and on three dimensional CR manifolds, Adv. in Math. 69 (1988), 233-303. MR 946264 (89g:32027)
  • 13. R. Fefferman Harmonic analysis on product spaces, Ann. of Math. 126 (1987), 109-130. MR 898053 (90e:42030)
  • 14. R. Fefferman and E.M. Stein, Singular integrals on product spaces, Adv. Math. 45 (1982), 117-143. MR 664621 (84d:42023)
  • 15. M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170. MR 1070037 (92a:46042)
  • 16. S. H. Ferguson and M. Lacey, A characterization of product BMO by commutators, Acta Math. 189 (2002), 143-160. MR 1961195 (2004e:42026)
  • 17. M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS-AMS Regional Conf. Ser. 79, 1991. MR 1107300 (92m:42021)
  • 18. G.B. Folland and E.M. Stein, Estimate for the $ \overline {\partial }_b$-complex and analysis on the Heisenberg group, Comm. Pure and Appl. Math. 27 (1974), 429-522. MR 0367477 (51:3719)
  • 19. Y. Guivarc'h, Groissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379. MR 0369608 (51:5841)
  • 20. R. Gundy and E.M. Stein, $ H^p$ theory for the polydisc, Proc. Nat. Acad. Sci. 76 (1979), 1026-1029. MR 524328 (80j:32012)
  • 21. Y.S. Han, Calderón-type reproducing formula and the Tb theorem, Rev. Mat. Iberoamericana 10 (1994), 51-91. MR 1271757 (95h:42020)
  • 22. Y. Han, Plancherel-Pôlya type inequality on space of homogeneous type and its applications, Proc. Amer. Math. Soc. 126 (1998), 3315-3327. MR 1459123 (99a:42010)
  • 23. Y. Han, Discrete Calderón-type reproducing formula, Acta Math. Sin. (Engl.Ser.), 16 (2000), 277-294. MR 1778708 (2001k:42024)
  • 24. Y. Han, J. Li and G. Lu, Duality of multiparameter Hardy space $ H^p$ on product spaces of homogeneous type, Ann. Sc. Nor. Super. Pisa Cl. Sc. (5) 9 (2010), no. 4, 645-685. MR 2789471
  • 25. Y. Han, J. Li, G. Lu and P.Y. Wang, $ H^p\rightarrow H^p$ boundedness implies $ H^p\rightarrow L^p$ boundedness, Forum Math. 23 (2011), no. 4, 729-756.
  • 26. Y. Han and G. Lu, Discrete Littlewood-Paley-Stein theory and multi-parameter Hardy spaces associated with the flag singular integrals, available at: http://arxiv.org/abs/0801.1701.
  • 27. Y. Han and G. Lu, Some Recent Works on Multiparameter Hardy Space Theory and Discrete Littlewood-Paley Analysis, Trends in Partial Differential Equations, ALM 10, 99-191, High Education Press and International Press (2009), Beijing-Boston. MR 2648281 (2011e:42043)
  • 28. Y. Han and G. Lu, Endpoint estimates for singular integral operators and Hardy spaces associated with Zygmund dilations, Preprint 2008.
  • 29. Y. Han, G. Lu and E. Sawyer, Implicit multiparameter Hardy spaces and Marcinkiewcz multipliers on the Heisenberg group, Preprint 2010.
  • 30. Y. Han, G. Lu and D. Yang Product theory on space of homogeneous type, unpublished manuscript.
  • 31. Y. Han, D. Müller and D. Yang, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstract and Applied Analysis, Vol 2008, Article ID 893409. MR 2485404 (2010c:46082)
  • 32. Y. Han, D. Müller and D. Yang, Littlewood-Paley-Stein characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), 1505-1537. MR 2269253 (2007g:42035)
  • 33. Y. Han and E. Sawyer, Littlewood-Paley-Stein theorem on space of homogeneous type and classical function spaces, Mem. Amer. Math. Soc. 110 (1994), no. 530, 1-126. MR 1214968 (96a:42016)
  • 34. B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Funda. Math. 25 (1935), 217-234.
  • 35. J.L. Journé, Calderón-Zygmund operators on product space, Rev. Mat. Iberoamericana 1 (1985), 55-92. MR 836284 (88d:42028)
  • 36. J.L. Journé, Two problems of Calderón-Zygmund theory on product spaces, Ann. Inst. Fourier (Grenoble) 38 (1988), 111-132. MR 949001 (90b:42031)
  • 37. K. Koenig, On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian, Amer. J. Math. 124 (2002), 129-197. MR 1879002 (2002m:32061)
  • 38. J.J. Kohn, Boundary behaviour of $ \overline {\partial }$ on weakly pseudoconvex manifolds of dimension two, J. Differential Geom. 6 (1972), 523-542. MR 0322365 (48:727)
  • 39. J.J. Kohn, Subellipticity of the $ \overline {\partial }$-Neumann problem on pseudoconvex domains: Sufficient conditions, Acta Math. 142 (1979), 79-122. MR 512213 (80d:32020)
  • 40. J.J. Kohn, Estimate for the $ \overline {\partial }_b$ on compact pseudoconvex CR manifolds, Proc. Symposia Pure Math. 43, 207-217, A.M.S., Providence, RI, 1985. MR 812292 (87c:32025)
  • 41. Y. Meyer, Wavelets and Operators, Cambridge University Press, 1992. MR 1228209 (94f:42001)
  • 42. R.A. Macías and C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), 271-309. MR 546296 (81c:32017b)
  • 43. D. Müller, F. Ricci and E.M. Stein, Marcinkiewicz multiplers and multiparameter structure on Heisenberg (-type) groups. I, Invent. Math. 119 (1995), 119-233. MR 1312498 (96b:43005)
  • 44. A. Nagel, F. Ricci and E.M. Stein, Harmonic analysis and the fundamental solutions on nilpotent Lie groups, in: Analysis and partial differential equations, 249-275, Lecture Notes in Pure and Appl. Math., 122, Dekker, New York, 1990. MR 1044793 (91m:22016)
  • 45. A. Nagel, J.P. Rosay, E.M. Stein and S. Wainger, Estimates for the Bergman and Szegö kernels in $ \mathbb{C}^2$, Ann. of Math. 129 (1989), 113-149. MR 979602 (90g:32028)
  • 46. A. Nagel and E.M. Stein, The $ \Box _b$-Heat equation on pseudoconvex manifolds of finite type in $ \mathbb{C}^2$, Math. Z. 238 (2001), 37-88. MR 1860735 (2002h:32031)
  • 47. A. Nagel and E.M. Stein, Differentiable control metrics and scaled bump functions, J. Differential Geom. 57 (2001), 37-88. MR 1882665 (2003i:58003)
  • 48. A. Nagel and E.M. Stein, The $ \bar {\partial }_b$-complex on decoupled boundarise in $ \mathbb{C}^n$, Ann. of Math. (2) 164 (2006), 649-713. MR 2247970 (2007d:32036)
  • 49. A. Nagel and E.M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoamericana 20 (2004), 531-561. MR 2073131 (2006i:42023)
  • 50. A. Nagel, E.M. Stein and S. Wainger, Balls and metrics defined by vector fields I. Basic properties, Acta Math. 155 (1985), 103-147. MR 793239 (86k:46049)
  • 51. J. Pipher, Journé's covering lemma and its extension to higher dimensions, Duke Mathematical Journal 53 (1986), 683-690. MR 860666 (88a:42019)
  • 52. L.P. Rothschild and E.M. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247-320. MR 0436223 (55:9171)
  • 53. E. Sawyer and R. Wheeden, Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114 (1992), 813-874. MR 1175693 (94i:42024)
  • 54. E.M. Stein, Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, NJ (1993). MR 1232192 (95c:42002)
  • 55. E.M. Stein, Some geometrical concepts arising in harmonic analysis, Geom. Func. Anal. 2000, Special Volume, Part I, 434-453. MR 1826263 (2002f:42014)

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Additional Information

Yongsheng Han
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
Email: hanyong@auburn.edu

Ji Li
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Email: liji6@mail.sysu.edu.cn

Guozhen Lu
Affiliation: School of Mathematical Science, Beijing Normal University, Beijing, 100875, People’s Republic of China – and – Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: gzlu@math.wayne.edu

DOI: https://doi.org/10.1090/S0002-9947-2012-05638-8
Keywords: Kohn-Laplacian, heat equation, Shilov boundary, finite type domains, multiparameter Hardy space, Carleson measure space, Calderón’s reproducing formula, almost orthogonality estimate, sup-inf-type inequality, sequence spaces, duality, singular integral operators.
Received by editor(s): May 13, 2010
Received by editor(s) in revised form: February 21, 2011
Published electronically: July 24, 2012
Additional Notes: The first two authors were supported by NNSF of China (Grant No. 11001275).
The third author is the corresponding author and was partly supported by US National Science Foundation grants DMS0500853 and DMS0901761.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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