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Classes of Hardy spaces associated with operators, duality theorem and applications


Author: Lixin Yan
Journal: Trans. Amer. Math. Soc. 360 (2008), 4383-4408
MSC (2000): Primary 42B30, 42B35, 47B38
DOI: https://doi.org/10.1090/S0002-9947-08-04476-0
Published electronically: March 20, 2008
MathSciNet review: 2395177
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Abstract: Let $ L$ be the infinitesimal generator of an analytic semigroup on $ L^2({\mathbb{R}}^n)$ with suitable upper bounds on its heat kernels. In Auscher, Duong, and McIntosh (2005) and Duong and Yan (2005), a Hardy space $ H^1_L({\mathbb{R}}^n)$ and a $ {\rm BMO}_L({\mathbb{R}}^n)$ space associated with the operator $ L$ were introduced and studied. In this paper we define a class of $ H^p_L({\mathbb{R}}^n)$ spaces associated with the operator $ L$ for a range of $ p<1$ acting on certain spaces of Morrey-Campanato functions defined in New Morrey-Campanato spaces associated with operators and applications by Duong and Yan (2005), and they generalize the classical $ H^p({\mathbb{R}}^n)$ spaces. We then establish a duality theorem between the $ H^p_L({\mathbb{R}}^n)$ spaces and the Morrey-Campanato spaces in that same paper. As applications, we obtain the boundedness of fractional integrals on $ H^p_L({\mathbb{R}}^n)$ and give the inclusion between the classical $ H^p({\mathbb{R}}^n)$ spaces and the $ H^p_L({\mathbb{R}}^n)$ spaces associated with operators.


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  • 1. D. Albrecht, X.T. Duong and A. McIntosh, Operator theory and harmonic analysis, Instructional Workshop on Analysis and Geometry (Proc. Centre Math. Analysis, 34, A.N.U., Canberra, 1996) pp 77-136. MR 1394696 (97e:47001)
  • 2. P. Auscher and E. Russ, Hardy spaces and divergence operators on strongly Lipschitz domain of $ {\mathbb{R}}^n$, J. Funct. Anal., 201 (2003), 148-184. MR 1986158 (2004c:42049)
  • 3. P. Auscher and P. Tchamitchian, Calcul fonctionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Institut Fourier (Grenoble), 45 (1995), 721-778. MR 1340951 (96f:35036)
  • 4. P. Auscher and P. Tchamitchian, Square root problem for divergence operators and related topics, Asterisque, 249, Soc. Math. France, 1998. MR 1651262 (2000c:47092)
  • 5. P. Auscher, X.T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished preprint, (2005).
  • 6. S. Blunck and P.C. Kunstmann, Weak type $ (p,p)$ estimates for Riesz transforms, Math. Z., 247 (2004), 137-148. MR 2054523 (2005f:35071)
  • 7. T. Coulhon and X.T. Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Pr $ \ddot{\rm u}$ss, Adv. Differential Equations, 5 (2000), 343-368. MR 1734546 (2001d:34087)
  • 8. D-C. Chang, S.G. Krantz and E.M. Stein, $ H^p$ theory on a smooth domain in $ {\mathbb{R}}^n$ and elliptic boundary value problems, J. Funct. Anal., 114 (1993), 286-347. MR 1223705 (94j:46032)
  • 9. R.R. Coifman, Y. Meyer and E.M. Stein, Un nouvel espace adapté a l'étude des opérateurs définis par des intégrales singuliéres, in ``Proc. Conf. Harmonic Analysis, Cortona'', Lecture Notes in Math. Vol. 992, pp, 1-15, Springer-Verlag, Berlin/New York, 1983. MR 729344 (85j:42032)
  • 10. R.R. Coifman, Y. Meyer and E.M. Stein, Some new functions and their applications to harmonic analysis, J. Funct. Analysis, 62 (1985), 304-315. MR 791851 (86i:46029)
  • 11. E.B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, 1989. MR 990239 (90e:35123)
  • 12. D.G. Deng, X.T. Duong, A. Sikora and L.X. Yan, Comparison of the classical BMO with the BMO spaces associated with operators and applications, to appear, Rev. Mat. Iberoamericana (2008).
  • 13. D.G. Deng, X.T. Duong and L.X. Yan, A characterization of the Morrey-Campanato spaces, Math. Z., 250 (2005), 641-655. MR 2179615 (2006g:42039)
  • 14. X.T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana, 15 (1999), 233-265. MR 1715407 (2001e:42017a)
  • 15. X.T. Duong, E.M. Ouhabaz and L.X. Yan, Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Arkiv för Matematik 44 (2006), 261-275. MR 2292721 (2008a:35224)
  • 16. P.L. Duren, B.W. Romberg and A.L. Shields, Linear functionals on $ H^p$ spaces with $ 0<p<1$, J. Reine Angew. Math., 238 (1969), 32-60. MR 0259579 (41:4217)
  • 17. X.T. Duong and L.X. Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., 58 (2005), 1375-1420. MR 2162784 (2006i:26012)
  • 18. X.T. Duong and L.X. Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc., 18 (2005), 943-973. MR 2163867 (2006d:42037)
  • 19. X.T. Duong and L.X. Yan, New Morrey-Campanato spaces associated with operators and applications, preprint, 2005.
  • 20. J. Dziubański and J. Zienkiewicz, $ H^p$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math., 98 (2003), 5-38. MR 2032068 (2004k:42038)
  • 21. C. Fefferman and E.M. Stein, $ H^p$ spaces of several variables, Acta Math., 129 (1972), 137-195. MR 0447953 (56:6263)
  • 22. F. John and L. Nirenberg, On functions of bounded mean oscillation. Comm. Pure Appl. Math., 14 (1961), 415-426. MR 0131498 (24:A1348)
  • 23. S. Janson, M.H. Taibleson, G. Weiss, Elementary characterizations of the Morrey-Campanato spaces, Lecture Notes in Math., 992 (1983), 101-114. MR 729349 (85k:46033)
  • 24. S. Hofmann and J.M. Martell, $ L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat., 47 (2003), 497-515. MR 2006497 (2004i:35067)
  • 25. J.M. Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math., 161(2004), 113-145. MR 2033231 (2005b:42016)
  • 26. A. McIntosh, Operators which have an $ H_{\infty}$-calculus, Miniconference on operator theory and partial differential equations (Proc. Centre Math. Analysis, ANU, Canberra 14, 1986), 210-231. MR 912940 (88k:47019)
  • 27. E.M. Ouhabaz, Analysis of heat equations on domains. London Math. Soc. Mono. 31, Princeton Univ. Press, (2004). MR 2124040 (2005m:35001)
  • 28. S. Semmes, Square function estimates and the $ T(b)$ theorem, Proc. Amer. Math. Soc., 110 (1990), 721-726. MR 1028049 (91h:42018)
  • 29. E.M. Stein, Singular integral and differentiability properties of functions, Princeton Univ. Press, 30, (1970). MR 0290095 (44:7280)
  • 30. E.M. Stein, Harmonic analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
  • 31. W.A. Strauss, Partial differential equations: An introduction. John Wiley & Sons, Inc., New York, 1992. MR 1159712 (92m:35001)
  • 32. E.M. Stein and G. Weiss, On the theory of harmonic functions of several variables I, The theory of $ H^p$ spaces, Acta Math., 103 (1960), 25-62. MR 0121579 (22:12315)
  • 33. A. Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Math., Vol 123, Academic Press, (1986). MR 869816 (88e:42001)
  • 34. M.W. Taibleson and G. Weiss, The molecular characterization of certain Hardy spaces, Ast$ \acute{e}$risque. 77, 68-149 (1980). MR 0604370 (83g:42012)
  • 35. A. Uchiyama and J.M. Wilson, Approximate identities and $ H^1({\mathbb{R}})$, Proc. Amer. Math. Soc., 88 (1983), 53-58. MR 691278 (84c:42032)
  • 36. N. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and geometry on groups. Cambridge Univ. Press, London, 1993. MR 1218884 (95f:43008)
  • 37. G. Weiss, Some problems in the theory of Hardy spaces, Proc. Sympos. Pure Math. Vol. 35, Amer. Math. Soc., Providence, R.I., (1979), 189-200. MR 545258 (80j:30051)
  • 38. L.X. Yan, Littlewood-Paley functions associated to second order elliptic operators, Math. Z., 246, (2004), 655-666. MR 2045834 (2005a:42015)
  • 39. K. Yosida, Functional Analysis (Fifth edition), Springer-Verlag, Berlin, 1978. MR 0500055 (58:17765)

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Additional Information

Lixin Yan
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Email: mcsylx@mail.sysu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-08-04476-0
Received by editor(s): July 15, 2005
Received by editor(s) in revised form: September 5, 2006
Published electronically: March 20, 2008
Additional Notes: The author was supported by NNSF of China (Grant No. 10571182/10771221) and by a grant from the Australia Research Council.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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