Exponential self-improvement of generalized Poincaré inequalities associated with approximations of the identity and semigroups
HTML articles powered by AMS MathViewer
- by Ana Jiménez-del-Toro PDF
- Trans. Amer. Math. Soc. 364 (2012), 637-660 Request permission
Abstract:
The purpose of this paper is to present a general method that allows us to study exponential self-improving properties of generalized Poincaré inequalities associated with an approximation of the identity or a semigroup. In particular, we show the connection between our results and the John-Nirenberg theorem for the space $BMO$ associated with approximations of the identity and semigroups.References
- Pascal Auscher, Thierry Coulhon, Xuan Thinh Duong, and Steve Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911–957 (English, with English and French summaries). MR 2119242, DOI 10.1016/j.ansens.2004.10.003
- Nadine Badr, Ana Jiménez-del-Toro, and José María Martell, $L^p$ self-improvement of generalized Poincaré inequalities in spaces of homogeneous type, J. Funct. Anal. 260 (2011), no. 11, 3147–3188. MR 2776565, DOI 10.1016/j.jfa.2011.01.014
- D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), no. 4, 1033–1074. MR 1386760, DOI 10.1512/iumj.1995.44.2019
- Colin Bennett and Robert Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Inc., Boston, MA, 1988. MR 928802
- D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304. MR 440695, DOI 10.1007/BF02394573
- Peter Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 2, 213–230. MR 683635
- 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
- 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
- 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 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
- Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
- Bruno Franchi, Inégalités de Sobolev pour des champs de vecteurs lipschitziens, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 6, 329–332 (French, with English summary). MR 1071637
- Bruno Franchi, Carlos Pérez, and Richard L. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), no. 1, 108–146. MR 1609261, DOI 10.1006/jfan.1997.3175
- José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
- Nicola Garofalo and Duy-Minh Nhieu, Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces, Comm. Pure Appl. Math. 49 (1996), no. 10, 1081–1144. MR 1404326, DOI 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A
- Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
- Piotr Hajłasz and Pekka Koskela, Sobolev meets Poincaré, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 10, 1211–1215 (English, with English and French summaries). MR 1336257
- Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
- Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
- David Jerison, The Poincaré inequality for vector fields satisfying Hörmander’s condition, Duke Math. J. 53 (1986), no. 2, 503–523. MR 850547, DOI 10.1215/S0012-7094-86-05329-9
- A. Jiménez-del-Toro and J.M. Martell, Self-improvement of Poincaré type inequalities associated with approximations of the identity and semigroups, Preprint, 2009.
- 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
- Guozhen Lu, Embedding theorems on Campanato-Morrey spaces for vector fields on Hörmander type, Approx. Theory Appl. (N.S.) 14 (1998), no. 1, 69–80. MR 1651473
- 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
- Paul MacManus and Carlos Pérez, Trudinger inequalities without derivatives, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1997–2012. MR 1881027, DOI 10.1090/S0002-9947-02-02918-5
- 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
- Louis Nirenberg, Estimates and existence of solutions of elliptic equations, Comm. Pure Appl. Math. 9 (1956), 509–529. MR 91402, DOI 10.1002/cpa.3160090322
- Lin Tang, New function spaces of Morrey-Campanato type on spaces of homogeneous type, Illinois J. Math. 51 (2007), no. 2, 625–644. MR 2342680
- M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 146, Marcel Dekker, Inc., New York, 1991. MR 1113700
- Laurent Saloff-Coste, Uniformly elliptic operators on Riemannian manifolds, J. Differential Geom. 36 (1992), no. 2, 417–450. MR 1180389
- L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 2 (1992), 27–38. MR 1150597, DOI 10.1155/S1073792892000047
- Laurent Saloff-Coste, Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, vol. 289, Cambridge University Press, Cambridge, 2002. MR 1872526
- Sven Spanne, Some function spaces defined using the mean oscillation over cubes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 19 (1965), 593–608. MR 190729
- 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
- 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
- V. I. Yudovich, Dokl. Akad. Nauk SSSR 138 (1961), 805–808 (English translation: Soviet Math., no. 2-3, 1961, 746–749).
Additional Information
- Ana Jiménez-del-Toro
- Affiliation: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain
- Email: anajimtor@hotmail.com
- Received by editor(s): March 13, 2009
- Received by editor(s) in revised form: July 16, 2009
- Published electronically: September 14, 2011
- Additional Notes: This research was supported by MEC Grant MTM2007-60952 and by UAM-CM Grant CCG07-UAM/ESP-1664.
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 637-660
- MSC (2000): Primary 46E35; Secondary 47D06, 46E30, 42B25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05344-4
- MathSciNet review: 2846346