Abstract and Applied Analysis

Abstract and Applied Analysis / 2008 / Article

Research Article | Open Access

Volume 2008 |Article ID 893409 | https://doi.org/10.1155/2008/893409

Yongsheng Han, Detlef Müller, Dachun 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, 250 pages, 2008. https://doi.org/10.1155/2008/893409

A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

Academic Editor: Stephen Clark
Received13 Feb 2008
Accepted23 May 2008
Published04 Mar 2009

Abstract

We work on RD-spaces 𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in 𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spaces Hp(𝒳) and local Hardy spaces hp(𝒳) on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces when p1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains in N. Our theory applies in a wide range of settings.

References

  1. H. Triebel, Theory of Function Spaces. II, vol. 84 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1992. View at: Zentralblatt MATH | MathSciNet
  2. H. Triebel, Theory of Function Spaces. III, vol. 100 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2006. View at: Zentralblatt MATH | MathSciNet
  3. H. Triebel, Theory of Function Spaces, vol. 78 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1983. View at: Zentralblatt MATH | MathSciNet
  4. A. Jonsson and H. Wallin, “Function spaces on subsets of n,” Mathematical Reports, vol. 2, no. 1, pp. 1–221, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. H. Triebel, Fractals and Spectra, vol. 91 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 1997. View at: MathSciNet
  6. H. Triebel, The Structure of Functions, vol. 97 of Monographs in Mathematics, Birkhäuser, Basel, Switzerland, 2001. View at: Zentralblatt MATH | MathSciNet
  7. K.-T. Sturm, “On the geometry of metric measure spaces. I,” Acta Mathematica, vol. 196, no. 1, pp. 65–131, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  8. K.-T. Sturm, “On the geometry of metric measure spaces. II,” Acta Mathematica, vol. 196, no. 1, pp. 133–177, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. J. Lott and C. Villani, “Ricci curvature for metric-measure spaces via optimal transport,” to appear in Annals of Mathematics. Second Series. View at: Google Scholar
  10. S. Semmes, “An introduction to analysis on metric spaces,” Notices of the American Mathematical Society, vol. 50, no. 4, pp. 438–443, 2003. View at: Google Scholar | MathSciNet
  11. N. J. Korevaar and R. M. Schoen, “Sobolev spaces and harmonic maps for metric space targets,” Communications in Analysis and Geometry, vol. 1, no. 3-4, pp. 561–659, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. P. Hajłasz and P. Koskela, “Sobolev met Poincaré,” Memoirs of the American Mathematical Society, vol. 145, no. 688, pp. 1–101, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext, Springer, New York, NY, USA, 2001. View at: Zentralblatt MATH | MathSciNet
  14. A. Grigor'yan, J. Hu, and K.-S. Lau, “Heat kernels on metric measure spaces and an application to semilinear elliptic equations,” Transactions of the American Mathematical Society, vol. 355, no. 5, pp. 2065–2095, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  15. A. Grigor'yan, “Heat kernels and function theory on metric measure spaces,” in Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), vol. 338 of Contemporary Mathematics, pp. 143–172, American Mathematical Society, Providence, RI, USA, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  16. S. Keith and X. Zhong, “The Poincaré inequality is an open ended condition,” Annals of Mathematics. Second Series, vol. 167, no. 2, pp. 575–599, 2008. View at: Google Scholar
  17. P. Hajłasz, “Sobolev spaces on an arbitrary metric space,” Potential Analysis, vol. 5, no. 4, pp. 403–415, 1996. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  18. P. Koskela and P. MacManus, “Quasiconformal mappings and Sobolev spaces,” Studia Mathematica, vol. 131, no. 1, pp. 1–17, 1998. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  19. B. Franchi, P. Hajłasz, and P. Koskela, “Definitions of Sobolev classes on metric spaces,” Annales de l'Institut Fourier, vol. 49, no. 6, pp. 1903–1924, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  20. A. E. Gatto and S. Vági, “On Sobolev spaces of fractional order and ϵ-families of operators on spaces of homogeneous type,” Studia Mathematica, vol. 133, no. 1, pp. 19–27, 1999. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. Y. Liu, G. Lu, and R. L. Wheeden, “Some equivalent definitions of high order Sobolev spaces on stratified groups and generalizations to metric spaces,” Mathematische Annalen, vol. 323, no. 1, pp. 157–174, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  22. N. Shanmugalingam, “Newtonian spaces: an extension of Sobolev spaces to metric measure spaces,” Revista Matemática Iberoamericana, vol. 16, no. 2, pp. 243–279, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  23. J. Heinonen, P. Koskela, N. Shanmugalingam, and J. T. Tyson, “Sobolev classes of Banach space-valued functions and quasiconformal mappings,” Journal d'Analyse Mathématique, vol. 85, pp. 87–139, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  24. S. Klainerman and I. Rodnianski, “A geometric approach to the Littlewood-Paley theory,” Geometric and Functional Analysis, vol. 16, no. 1, pp. 126–163, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  25. D. Danielli, N. Garofalo, and D.-M. Nhieu, “Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces,” Memoirs of the American Mathematical Society, vol. 182, no. 857, pp. 1–119, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  26. P. Hajłasz, T. Iwaniec, J. Malý, and J. Onninen, “Weakly differentiable mappings between manifolds,” Memoirs of the American Mathematical Society, vol. 192, no. 899, pp. 1–72, 2008. View at: Google Scholar | MathSciNet
  27. G. Furioli, C. Melzi, and A. Veneruso, “Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth,” Mathematische Nachrichten, vol. 279, no. 9-10, pp. 1028–1040, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  28. R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis,” Bulletin of the American Mathematical Society, vol. 83, no. 4, pp. 569–645, 1977. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  29. R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes, vol. 242 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1971. View at: Zentralblatt MATH | MathSciNet
  30. R. A. Macías and C. Segovia, “A decomposition into atoms of distributions on spaces of homogeneous type,” Advances in Mathematics, vol. 33, no. 3, pp. 271–309, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  31. Y. Han, “Calderón-type reproducing formula and the Tb theorem,” Revista Matemática Iberoamericana, vol. 10, no. 1, pp. 51–91, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  32. X. T. Duong and L. Yan, “Hardy spaces of spaces of homogeneous type,” Proceedings of the American Mathematical Society, vol. 131, no. 10, pp. 3181–3189, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  33. R. A. Macías and C. Segovia, “Lipschitz functions on spaces of homogeneous type,” Advances in Mathematics, vol. 33, no. 3, pp. 257–270, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  34. Y. Han and E. T. Sawyer, “Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces,” Memoirs of the American Mathematical Society, vol. 110, no. 530, pp. 1–126, 1994. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  35. Y. Han, “Inhomogeneous Calderón reproducing formula on spaces of homogeneous type,” The Journal of Geometric Analysis, vol. 7, no. 2, pp. 259–284, 1997. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  36. Y. Han and D. Yang, “New characterizations and applications of inhomogeneous Besov and Triebel-Lizorkin spaces on homogeneous type spaces and fractals,” Dissertationes Mathematicae, vol. 403, pp. 1–102, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  37. Y. Han and D. Yang, “Some new spaces of Besov and Triebel-Lizorkin type on homogeneous spaces,” Studia Mathematica, vol. 156, no. 1, pp. 67–97, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  38. A. Bellaïche and J.-J. Risler, Eds., Sub-Riemannian Geometry, A. Bellaïche and J.-J. Risler, Eds., vol. 144 of Progress in Mathematics, Birkhäuser, Basel, Switzerland, 1996. View at: Zentralblatt MATH | MathSciNet
  39. R. S. Strichartz, “Sub-Riemannian geometry,” Journal of Differential Geometry, vol. 24, no. 2, pp. 221–263, 1986. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  40. E. M. Stein, “Some geometrical concepts arising in harmonic analysis,” Geometric and Functional Analysis, pp. 434–453, 2000, Special Volume, Part I. View at: Google Scholar | MathSciNet
  41. A. Nagel and E. M. Stein, “The b-heat equation on pseudoconvex manifolds of finite type in 2,” Mathematische Zeitschrift, vol. 238, no. 1, pp. 37–88, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  42. A. Nagel and E. M. Stein, “Differentiable control metrics and scaled bump functions,” Journal of Differential Geometry, vol. 57, no. 3, pp. 465–492, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  43. A. Nagel and E. M. Stein, “The ¯b -complex on decoupled boundaries in n,” Annals of Mathematics. Second Series, vol. 164, no. 2, pp. 649–713, 2006. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  44. A. Nagel and E. M. Stein, “On the product theory of singular integrals,” Revista Matemática Iberoamericana, vol. 20, no. 2, pp. 531–561, 2004. View at: Google Scholar | MathSciNet
  45. A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, “Estimates for the Bergman and Szegö kernels in 2,” Annals of Mathematics. Second Series, vol. 129, no. 1, pp. 113–149, 1989. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  46. D.-C. Chang, A. Nagel, and E. M. Stein, “Estimates for the ¯-Neumann problem in pseudoconvex domains of finite type in 2,” Acta Mathematica, vol. 169, no. 1, pp. 153–228, 1992. View at: Publisher Site | Google Scholar | MathSciNet
  47. K. D. Koenig, “On maximal Sobolev and Hölder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian,” American Journal of Mathematics, vol. 124, no. 1, pp. 129–197, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  48. Y. Han, D. Müller, and D. Yang, “Littlewood-Paley characterizations for Hardy spaces on spaces of homogeneous type,” Mathematische Nachrichten, vol. 279, no. 13-14, pp. 1505–1537, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  49. D. Müller and D. Yang, “A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces,” to appear in Forum Mathematicum. View at: Google Scholar
  50. J.-O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, vol. 1381 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1989. View at: Zentralblatt MATH | MathSciNet
  51. R. L. Wheeden, “A characterization of some weighted norm inequalities for the fractional maximal function,” Studia Mathematica, vol. 107, no. 3, pp. 257–272, 1993. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  52. A. Jonsson, “Besov spaces on closed subsets of n,” Transactions of the American Mathematical Society, vol. 341, no. 1, pp. 355–370, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  53. A. Jonsson, “Besov spaces on closed sets by means of atomic decompositions,” Research Reports, Department of Mathematics, University of Umeå, Umeå, Sweden, 1993. View at: Google Scholar
  54. P. Bylund and J. Gudayol, “On the existence of doubling measures with certain regularity properties,” Proceedings of the American Mathematical Society, vol. 128, no. 11, pp. 3317–3327, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  55. H. Federer, Geometric Measure Theory, vol. 153 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1969. View at: Zentralblatt MATH | MathSciNet
  56. A. Nagel, E. M. Stein, and S. Wainger, “Balls and metrics defined by vector fields. I. Basic properties,” Acta Mathematica, vol. 155, no. 1-2, pp. 103–147, 1985. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  57. N. Th. Varopoulos, “Analysis on Lie groups,” Journal of Functional Analysis, vol. 76, no. 2, pp. 346–410, 1988. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  58. N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, vol. 100 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1992. View at: Zentralblatt MATH | MathSciNet
  59. A. Nagel, F. Ricci, and E. M. Stein, “Harmonic analysis and fundamental solutions on nilpotent Lie groups,” in Analysis and Partial Differential Equations, vol. 122 of Lecture Notes in Pure and Applied Mathematics, pp. 249–275, Dekker, New York, NY, USA, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  60. G. Alexopoulos, “Spectral multipliers on Lie groups of polynomial growth,” Proceedings of the American Mathematical Society, vol. 120, no. 3, pp. 973–979, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  61. Y. Guivarc'h, “Croissance polynomiale et périodes des fonctions harmoniques,” Bulletin de la Société Mathématique de France, vol. 101, pp. 333–379, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  62. J. W. Jenkins, “Growth of connected locally compact groups,” Journal of Functional Analysis, vol. 12, pp. 113–127, 1973. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  63. A. Nagel and E. M. Stein, “Corrigenda: “On the product theory of singular integrals”,” Revista Matemática Iberoamericana, vol. 21, no. 2, pp. 693–694, 2005. View at: Google Scholar | MathSciNet
  64. G. David, J.-L. Journé, and S. Semmes, “Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation,” Revista Matemática Iberoamericana, vol. 1, no. 4, pp. 1–56, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  65. R. A. Macías, C. Segovia, and J. L. Torrea, “Singular integral operators with non-necessarily bounded kernels on spaces of homogeneous type,” Advances in Mathematics, vol. 93, no. 1, pp. 25–60, 1992. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  66. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, no. 30, Princeton University Press, Princeton, NJ, USA, 1970. View at: Zentralblatt MATH | MathSciNet
  67. K. Yosida, Functional Analysis, Classics in Mathematics, Springer, Berlin, Germany, 1995. View at: MathSciNet
  68. G. David and J.-L. Journé, “A boundedness criterion for generalized Calderón-Zygmund operators,” Annals of Mathematics. Second Series, vol. 120, no. 2, pp. 371–397, 1984. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  69. Y. Meyer, “Les nouveaux opérateurs de Calderón-Zygmund,” Astérisque, no. 131, pp. 237–254, 1985. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  70. F. John and L. Nirenberg, “On functions of bounded mean oscillation,” Communications on Pure and Applied Mathematics, vol. 14, no. 3, pp. 415–426, 1961. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  71. Y. Meyer and R. Coifman, Wavelets, vol. 48 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1997. View at: Zentralblatt MATH | MathSciNet
  72. M. Christ, “A T(b) theorem with remarks on analytic capacity and the Cauchy integral,” Colloquium Mathematicum, vol. 60/61, no. 2, pp. 601–628, 1990. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  73. D. Goldberg, “A local version of real Hardy spaces,” Duke Mathematical Journal, vol. 46, no. 1, pp. 27–42, 1979. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  74. Da. Yang, Do. Yang, and Y. Zhou, “Localized Campanato spaces on RD-spaces and their applications to Schrödinger operators,” submitted. View at: Google Scholar
  75. E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993. View at: Zentralblatt MATH | MathSciNet
  76. Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1992. View at: Zentralblatt MATH | MathSciNet
  77. C. Fefferman and E. M. Stein, “Some maximal inequalities,” American Journal of Mathematics, vol. 93, no. 1, pp. 107–115, 1971. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  78. M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, vol. 79 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1991. View at: Zentralblatt MATH | MathSciNet
  79. Y. Han, “Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications,” Proceedings of the American Mathematical Society, vol. 126, no. 11, pp. 3315–3327, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  80. D. Deng, Y. Han, and D. Yang, “Inhomogeneous Plancherel-Pôlya inequalities on spaces of homogeneous type and their applications,” Communications in Contemporary Mathematics, vol. 6, no. 2, pp. 221–243, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  81. J. Duoandikoetxea, Fourier Analysis, vol. 29 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 2001. View at: Zentralblatt MATH | MathSciNet
  82. M. Frazier and B. Jawerth, “A discrete transform and decompositions of distribution spaces,” Journal of Functional Analysis, vol. 93, no. 1, pp. 34–170, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  83. D. Yang, “Some new Triebel-Lizorkin spaces on spaces of homogeneous type and their frame characterizations,” Science in China. Series A, vol. 48, no. 1, pp. 12–39, 2005. View at: Google Scholar | MathSciNet
  84. D. Yang, “Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations,” Studia Mathematica, vol. 167, no. 1, pp. 63–98, 2005. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  85. X. T. Duong and L. Yan, “New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications,” Communications on Pure and Applied Mathematics, vol. 58, no. 10, pp. 1375–1420, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  86. Y. Han and D. Yang, “New characterization of BMO (n) space,” Boletín de la Sociedad Matemática Mexicanae. Tercera Serie, vol. 10, no. 1, pp. 95–103, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  87. H. Triebel, “Function spaces in Lipschitz domains and on Lipschitz manifolds. Characteristic functions as pointwise multipliers,” Revista Matemática Complutense, vol. 15, no. 2, pp. 475–524, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  88. J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, Germany, 1976. View at: Zentralblatt MATH | MathSciNet
  89. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Johann Ambrosius Barth, Heidelberg, Germany, 2nd edition, 1995. View at: Zentralblatt MATH | MathSciNet
  90. D. Yang, “Frame characterizations of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type and their applications,” Georgian Mathematical Journal, vol. 9, no. 3, pp. 567–590, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  91. D. Yang, “Real interpolations for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type,” Mathematische Nachrichten, vol. 273, no. 1, pp. 96–113, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

Copyright © 2008 Yongsheng Han et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views1150
Downloads4937
Citations

Related articles