Classes of Hardy spaces associated with operators, duality theorem and applications
HTML articles powered by AMS MathViewer
- by Lixin Yan PDF
- Trans. Amer. Math. Soc. 360 (2008), 4383-4408 Request permission
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 $\textrm {BMO}_L({\mathbb R}^n)$ 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.References
- David Albrecht, Xuan Duong, and Alan McIntosh, Operator theory and harmonic analysis, Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 34, Austral. Nat. Univ., Canberra, 1996, pp. 77–136. MR 1394696
- Pascal Auscher and Emmanuel Russ, Hardy spaces and divergence operators on strongly Lipschitz domains of $\Bbb R^n$, J. Funct. Anal. 201 (2003), no. 1, 148–184. MR 1986158, DOI 10.1016/S0022-1236(03)00059-4
- P. Auscher and P. Tchamitchian, Calcul fontionnel précisé pour des opérateurs elliptiques complexes en dimension un (et applications à certaines équations elliptiques complexes en dimension deux), Ann. Inst. Fourier (Grenoble) 45 (1995), no. 3, 721–778 (French, with English and French summaries). MR 1340951
- Pascal Auscher and Philippe Tchamitchian, Square root problem for divergence operators and related topics, Astérisque 249 (1998), viii+172 (English, with English and French summaries). MR 1651262
- P. Auscher, X.T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, unpublished preprint, (2005).
- S. Blunck and P. C. Kunstmann, Weak type $(p,p)$ estimates for Riesz transforms, Math. Z. 247 (2004), no. 1, 137–148. MR 2054523, DOI 10.1007/s00209-003-0627-7
- Thierry Coulhon and Xuan Thinh Duong, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations 5 (2000), no. 1-3, 343–368. MR 1734546
- Der-Chen Chang, Steven G. Krantz, and Elias M. Stein, $H^p$ theory on a smooth domain in $\textbf {R}^N$ and elliptic boundary value problems, J. Funct. Anal. 114 (1993), no. 2, 286–347. MR 1223705, DOI 10.1006/jfan.1993.1069
- R. R. Coifman, Y. Meyer, and E. M. Stein, Un nouvel éspace fonctionnel adapté à l’étude des opérateurs définis par des intégrales singulières, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 1–15 (French). MR 729344, DOI 10.1007/BFb0069149
- R. R. Coifman, Y. Meyer, and E. M. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), no. 2, 304–335. MR 791851, DOI 10.1016/0022-1236(85)90007-2
- E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
- 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) .
- Donggao Deng, Xuan Thinh Duong, and Lixin Yan, A characterization of the Morrey-Campanato spaces, Math. Z. 250 (2005), no. 3, 641–655. MR 2179615, DOI 10.1007/s00209-005-0769-x
- Xuan Thinh Duong and Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233–265. MR 1715407, DOI 10.4171/RMI/255
- Xuan Thinh Duong, El Maati Ouhabaz, and Lixin Yan, Endpoint estimates for Riesz transforms of magnetic Schrödinger operators, Ark. Mat. 44 (2006), no. 2, 261–275. MR 2292721, DOI 10.1007/s11512-006-0021-x
- 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 259579
- Xuan Thinh Duong and Lixin Yan, New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications, Comm. Pure Appl. Math. 58 (2005), no. 10, 1375–1420. MR 2162784, DOI 10.1002/cpa.20080
- Xuan Thinh Duong and Lixin Yan, Duality of Hardy and BMO spaces associated with operators with heat kernel bounds, J. Amer. Math. Soc. 18 (2005), no. 4, 943–973. MR 2163867, DOI 10.1090/S0894-0347-05-00496-0
- X.T. Duong and L.X. Yan, New Morrey-Campanato spaces associated with operators and applications, preprint, 2005.
- Jacek Dziubański and Jacek Zienkiewicz, $H^p$ spaces associated with Schrödinger operators with potentials from reverse Hölder classes, Colloq. Math. 98 (2003), no. 1, 5–38. MR 2032068, DOI 10.4064/cm98-1-2
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 447953, DOI 10.1007/BF02392215
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
- Svante Janson, Mitchell Taibleson, and Guido Weiss, Elementary characterizations of the Morrey-Campanato spaces, Harmonic analysis (Cortona, 1982) Lecture Notes in Math., vol. 992, Springer, Berlin, 1983, pp. 101–114. MR 729349, DOI 10.1007/BFb0069154
- Steve Hofmann and José María Martell, $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), no. 2, 497–515. MR 2006497, DOI 10.5565/PUBLMAT_{4}7203_{1}2
- José María Martell, Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2004), no. 2, 113–145. MR 2033231, DOI 10.4064/sm161-2-2
- Alan McIntosh, Operators which have an $H_\infty$ functional calculus, Miniconference on operator theory and partial differential equations (North Ryde, 1986) Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 14, Austral. Nat. Univ., Canberra, 1986, pp. 210–231. MR 912940
- El Maati Ouhabaz, Analysis of heat equations on domains, London Mathematical Society Monographs Series, vol. 31, Princeton University Press, Princeton, NJ, 2005. MR 2124040
- Stephen Semmes, Square function estimates and the $T(b)$ theorem, Proc. Amer. Math. Soc. 110 (1990), no. 3, 721–726. MR 1028049, DOI 10.1090/S0002-9939-1990-1028049-2
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Walter A. Strauss, Partial differential equations, John Wiley & Sons, Inc., New York, 1992. An introduction. MR 1159712
- Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of $H^{p}$-spaces, Acta Math. 103 (1960), 25–62. MR 121579, DOI 10.1007/BF02546524
- Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
- Mitchell H. Taibleson and Guido Weiss, The molecular characterization of certain Hardy spaces, Representation theorems for Hardy spaces, Astérisque, vol. 77, Soc. Math. France, Paris, 1980, pp. 67–149. MR 604370
- Akihito Uchiyama and J. Michael Wilson, Approximate identities and $H^{1}(\textbf {R})$, Proc. Amer. Math. Soc. 88 (1983), no. 1, 53–58. MR 691278, DOI 10.1090/S0002-9939-1983-0691278-8
- N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and geometry on groups, Cambridge Tracts in Mathematics, vol. 100, Cambridge University Press, Cambridge, 1992. MR 1218884
- Guido Weiss, Some problems in the theory of Hardy spaces, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 189–200. MR 545258
- Lixin Yan, Littlewood-Paley functions associated to second order elliptic operators, Math. Z. 246 (2004), no. 4, 655–666. MR 2045834, DOI 10.1007/s00209-003-0606-z
- Kôsaku Yosida, Functional analysis, 5th ed., Grundlehren der Mathematischen Wissenschaften, Band 123, Springer-Verlag, Berlin-New York, 1978. MR 0500055
Additional Information
- Lixin Yan
- Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
- MR Author ID: 618148
- Email: mcsylx@mail.sysu.edu.cn
- 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.
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2395177