Localized Hardy spaces $H^1$ related to admissible functions on RD-spaces and applications to Schrödinger operators
HTML articles powered by AMS MathViewer
- by Dachun Yang and Yuan Zhou PDF
- Trans. Amer. Math. Soc. 363 (2011), 1197-1239 Request permission
Abstract:
Let ${\mathcal X}$ be an RD-space, which means that ${\mathcal X}$ is a space of homogenous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in ${\mathcal X}$. In this paper, the authors first introduce the notion of admissible functions $\rho$ and then develop a theory of localized Hardy spaces $H^1_\rho ({\mathcal X})$ associated with $\rho$, which includes several maximal function characterizations of $H^1_\rho ({\mathcal X})$, the relations between $H^1_\rho ({\mathcal X})$ and the classical Hardy space $H^1({\mathcal X})$ via constructing a kernel function related to $\rho$, the atomic decomposition characterization of $H^1_\rho ({\mathcal X})$, and the boundedness of certain localized singular integrals on $H^1_\rho ({\mathcal X})$ via a finite atomic decomposition characterization of some dense subspace of $H^1_\rho ({\mathcal X})$. This theory has a wide range of applications. Even when this theory is applied, respectively, to the Schrödinger operator or the degenerate Schrödinger operator on $\mathbb {R}^n$, or to the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups, some new results are also obtained. The Schrödinger operators considered here are associated with nonnegative potentials satisfying the reverse Hölder inequality.References
- G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), no. 4, 691–727 (English, with English and French summaries). MR 1178564, DOI 10.4153/CJM-1992-042-x
- Pascal Auscher and Besma Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 6, 1975–2013 (English, with English and French summaries). MR 2377893, DOI 10.5802/aif.2320
- Marcin Bownik, Boundedness of operators on Hardy spaces via atomic decompositions, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3535–3542. MR 2163588, DOI 10.1090/S0002-9939-05-07892-5
- Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
- R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
- Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948, DOI 10.1007/BFb0058946
- Ronald R. Coifman and Guido Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), no. 4, 569–645. MR 447954, DOI 10.1090/S0002-9904-1977-14325-5
- G. David, J.-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iberoamericana 1 (1985), no. 4, 1–56 (French). MR 850408, DOI 10.4171/RMI/17
- 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
- Jacek Dziubański, Atomic decomposition of $H^p$ spaces associated with some Schrödinger operators, Indiana Univ. Math. J. 47 (1998), no. 1, 75–98. MR 1631616, DOI 10.1512/iumj.1998.47.1479
- Jacek Dziubański and Jacek Zienkiewicz, Hardy space $H^1$ associated to Schrödinger operator with potential satisfying reverse Hölder inequality, Rev. Mat. Iberoamericana 15 (1999), no. 2, 279–296. MR 1715409, DOI 10.4171/RMI/257
- Jacek Dziubański and Jacek Zienkiewicz, $H^p$ spaces for Schrödinger operators, Fourier analysis and related topics (Będlewo, 2000) Banach Center Publ., vol. 56, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 45–53. MR 1971563, DOI 10.4064/bc56-0-4
- 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
- Jacek Dziubański and Jacek Zienkiewicz, Hardy spaces $H^1$ for Schrödinger operators with certain potentials, Studia Math. 164 (2004), no. 1, 39–53. MR 2079769, DOI 10.4064/sm164-1-3
- Jacek Dziubański, Note on $H^1$ spaces related to degenerate Schrödinger operators, Illinois J. Math. 49 (2005), no. 4, 1271–1297. MR 2210363
- J. Dziubański, G. Garrigós, T. Martínez, J. L. Torrea, and J. Zienkiewicz, $BMO$ spaces related to Schrödinger operators with potentials satisfying a reverse Hölder inequality, Math. Z. 249 (2005), no. 2, 329–356. MR 2115447, DOI 10.1007/s00209-004-0701-9
- Charles L. Fefferman, The uncertainty principle, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 2, 129–206. MR 707957, DOI 10.1090/S0273-0979-1983-15154-6
- 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
- G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581
- Loukas Grafakos, Modern Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 250, Springer, New York, 2009. MR 2463316, DOI 10.1007/978-0-387-09434-2
- Loukas Grafakos, LiGuang Liu, and DaChun Yang, Maximal function characterizations of Hardy spaces on RD-spaces and their applications, Sci. China Ser. A 51 (2008), no. 12, 2253–2284. MR 2462027, DOI 10.1007/s11425-008-0057-4
- Loukas Grafakos, Liguang Liu, and Dachun Yang, Radial maximal function characterizations for Hardy spaces on RD-spaces, Bull. Soc. Math. France 137 (2009), no. 2, 225–251 (English, with English and French summaries). MR 2543475, DOI 10.24033/bsmf.2574
- F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
- David Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27–42. MR 523600
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Yongsheng Han, Detlef Müller, and Dachun Yang, Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type, Math. Nachr. 279 (2006), no. 13-14, 1505–1537. MR 2269253, DOI 10.1002/mana.200610435
- Yongsheng Han, Detlef Müller, and Dachun Yang, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces, Abstr. Appl. Anal. , posted on (2008), Art. ID 893409, 250. MR 2485404, DOI 10.1155/2008/893409
- W. Hebisch and L. Saloff-Coste, On the relation between elliptic and parabolic Harnack inequalities, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 5, 1437–1481 (English, with English and French summaries). MR 1860672, DOI 10.5802/aif.1861
- Tosio Kato, Trotter’s product formula for an arbitrary pair of self-adjoint contraction semigroups, Topics in functional analysis (essays dedicated to M. G. Kreĭn on the occasion of his 70th birthday), Adv. in Math. Suppl. Stud., vol. 3, Academic Press, New York-London, 1978, pp. 185–195. MR 538020
- Kazuhiro Kurata and Satoko Sugano, Fundamental solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptic operators with positive potentials, Studia Math. 138 (2000), no. 2, 101–119. MR 1749075
- Kazuhiro Kurata and Satoko Sugano, A remark on estimates for uniformly elliptic operators on weighted $L^p$ spaces and Morrey spaces, Math. Nachr. 209 (2000), 137–150. MR 1734362, DOI 10.1002/(SICI)1522-2616(200001)209:1<137::AID-MANA137>3.3.CO;2-V
- Noël Lohoué and Nicolas Th. Varopoulos, Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 11, 559–560 (French, with English summary). MR 816628
- Hong-Quan Li, Estimations $L^p$ des opérateurs de Schrödinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), no. 1, 152–218 (French, with French summary). MR 1670222, DOI 10.1006/jfan.1998.3347
- C. Lin, H. Liu and Y. Liu, The Hardy space $H^1_L$ associated with Schrödinger operators on the Heisenberg group, Submitted.
- Guozhen Lu, A Fefferman-Phong type inequality for degenerate vector fields and applications, PanAmer. Math. J. 6 (1996), no. 4, 37–57. MR 1414658
- Roberto A. Macías and Carlos Segovia, Lipschitz functions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 257–270. MR 546295, DOI 10.1016/0001-8708(79)90012-4
- Roberto A. Macías and Carlos Segovia, A decomposition into atoms of distributions on spaces of homogeneous type, Adv. in Math. 33 (1979), no. 3, 271–309. MR 546296, DOI 10.1016/0001-8708(79)90013-6
- Stefano Meda, Peter Sjögren, and Maria Vallarino, On the $H^1$-$L^1$ boundedness of operators, Proc. Amer. Math. Soc. 136 (2008), no. 8, 2921–2931. MR 2399059, DOI 10.1090/S0002-9939-08-09365-9
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- Alexander Nagel and Elias M. Stein, On the product theory of singular integrals, Rev. Mat. Iberoamericana 20 (2004), no. 2, 531–561. MR 2073131, DOI 10.4171/RMI/400
- Alexander Nagel and Elias M. Stein, The $\overline {\partial }_b$-complex on decoupled boundaries in $\Bbb C^n$, Ann. of Math. (2) 164 (2006), no. 2, 649–713. MR 2247970, DOI 10.4007/annals.2006.164.649
- Alexander Nagel, Elias M. Stein, and Stephen Wainger, Balls and metrics defined by vector fields. I. Basic properties, Acta Math. 155 (1985), no. 1-2, 103–147. MR 793239, DOI 10.1007/BF02392539
- Laurent Saloff-Coste, Analyse sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math. 302 (1986), no. 14, 499–502 (French, with English summary). MR 844151
- Laurent Saloff-Coste, Fonctions maximales sur certains groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 305 (1987), no. 11, 457–459 (French, with English summary). MR 916309
- Laurent Saloff-Coste, Analyse sur les groupes de Lie à croissance polynômiale, Ark. Mat. 28 (1990), no. 2, 315–331 (French, with English summary). MR 1084020, DOI 10.1007/BF02387385
- Zhong Wei Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), no. 2, 513–546 (English, with English and French summaries). MR 1343560, DOI 10.5802/aif.1463
- 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
- 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
- Jan-Olov Strömberg and Alberto Torchinsky, Weighted Hardy spaces, Lecture Notes in Mathematics, vol. 1381, Springer-Verlag, Berlin, 1989. MR 1011673, DOI 10.1007/BFb0091154
- Hans Triebel, Theory of function spaces. III, Monographs in Mathematics, vol. 100, Birkhäuser Verlag, Basel, 2006. MR 2250142
- Akihito Uchiyama, A maximal function characterization of $H^{p}$ on the space of homogeneous type, Trans. Amer. Math. Soc. 262 (1980), no. 2, 579–592. MR 586737, DOI 10.1090/S0002-9947-1980-0586737-4
- N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), no. 2, 346–410. MR 924464, DOI 10.1016/0022-1236(88)90041-9
- 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
- P. Wojtaszczyk, Banach spaces for analysts, Cambridge Studies in Advanced Mathematics, vol. 25, Cambridge University Press, Cambridge, 1991. MR 1144277, DOI 10.1017/CBO9780511608735
- Dachun Yang and Yuan Zhou, A boundedness criterion via atoms for linear operators in Hardy spaces, Constr. Approx. 29 (2009), no. 2, 207–218. MR 2481589, DOI 10.1007/s00365-008-9015-1
- D. Yang and Y. Zhou, New properties of Besov and Triebel-Lizorkin spaces on RD-spaces, Manuscripta Math. (to appear), DOI: 10.1007/s00229-010-0384-y.
- Jiaping Zhong, The Sobolev estimates for some Schrödinger type operators, Math. Sci. Res. Hot-Line 3 (1999), no. 8, 1–48. MR 1717791
Additional Information
- Dachun Yang
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
- MR Author ID: 317762
- Email: dcyang@bnu.edu.cn
- Yuan Zhou
- Affiliation: School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, People’s Republic of China
- Email: yuanzhou@mail.bnu.edu.cn
- Received by editor(s): August 26, 2008
- Published electronically: October 22, 2010
- Additional Notes: The first author was supported by the National Natural Science Foundation (Grant No. 10871025) of China.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1197-1239
- MSC (2010): Primary 42B30; Secondary 42B20, 42B25, 42B35, 42B37
- DOI: https://doi.org/10.1090/S0002-9947-2010-05201-8
- MathSciNet review: 2737263