Abstract and Applied Analysis

Volume 2008 (2008), Article ID 893409, 250 pages

http://dx.doi.org/10.1155/2008/893409

Research Article

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

^{1}Department of Mathematics, Auburn University, Auburn, AL 36849-5310, USA^{2}Mathematisches Seminar, Christian-Albrechts-Universität Kiel, Ludewig-Meyn Strasse 4, 24098 Kiel, Germany^{3}School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

Received 13 February 2008; Accepted 23 May 2008

Academic Editor: Stephen Clark

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.

#### Linked References

- H. Triebel,
*Theory of Function Spaces. II*, vol. 84 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 1992. View at Zentralblatt MATH · View at MathSciNet - H. Triebel,
*Theory of Function Spaces. III*, vol. 100 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 2006. View at Zentralblatt MATH · View at MathSciNet - H. Triebel,
*Theory of Function Spaces*, vol. 78 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 1983. View at Zentralblatt MATH · View at MathSciNet - A. Jonsson and H. Wallin, “Function spaces on subsets of ${\mathbb{R}}^{n}$,”
*Mathematical Reports*, vol. 2, no. 1, pp. 1–221, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Triebel,
*Fractals and Spectra*, vol. 91 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 1997. View at MathSciNet - H. Triebel,
*The Structure of Functions*, vol. 97 of*Monographs in Mathematics*, Birkhäuser, Basel, Switzerland, 2001. View at Zentralblatt MATH · View at MathSciNet - K.-T. Sturm, “On the geometry of metric measure spaces. I,”
*Acta Mathematica*, vol. 196, no. 1, pp. 65–131, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K.-T. Sturm, “On the geometry of metric measure spaces. II,”
*Acta Mathematica*, vol. 196, no. 1, pp. 133–177, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lott and C. Villani, “Ricci curvature for metric-measure spaces via optimal transport,” to appear in
*Annals of Mathematics. Second Series*. - 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 · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - J. Heinonen,
*Lectures on Analysis on Metric Spaces*, Universitext, Springer, New York, NY, USA, 2001. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 - P. Hajłasz, “Sobolev spaces on an arbitrary metric space,”
*Potential Analysis*, vol. 5, no. 4, pp. 403–415, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Koskela and P. MacManus, “Quasiconformal mappings and Sobolev spaces,”
*Studia Mathematica*, vol. 131, no. 1, pp. 1–17, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - A. E. Gatto and S. Vági, “On Sobolev spaces of fractional order and $\u03f5$-families of operators on spaces of homogeneous type,”
*Studia Mathematica*, vol. 133, no. 1, pp. 19–27, 1999. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - R. S. Strichartz, “Sub-Riemannian geometry,”
*Journal of Differential Geometry*, vol. 24, no. 2, pp. 221–263, 1986. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - A. Nagel and E. M. Stein, “The ${\square}_{b}$-heat equation on pseudoconvex manifolds of finite type in ${\u2102}^{2}$,”
*Mathematische Zeitschrift*, vol. 238, no. 1, pp. 37–88, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - A. Nagel and E. M. Stein, “The ${\overline{\partial}}_{b}$
-complex on decoupled boundaries in ${\u2102}^{n}$,”
*Annals of Mathematics. Second Series*, vol. 164, no. 2, pp. 649–713, 2006. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - A. Nagel, J.-P. Rosay, E. M. Stein, and S. Wainger, “Estimates for the Bergman and Szegö kernels in ${\u2102}^{2}$,”
*Annals of Mathematics. Second Series*, vol. 129, no. 1, pp. 113–149, 1989. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D.-C. Chang, A. Nagel, and E. M. Stein, “Estimates for the $\overline{\partial}$-Neumann problem in pseudoconvex domains of finite type in ${\u2102}^{2}$,”
*Acta Mathematica*, vol. 169, no. 1, pp. 153–228, 1992. View at Publisher · View at Google Scholar · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at MathSciNet - D. Müller and D. Yang, “A difference characterization of Besov and Triebel-Lizorkin spaces on RD-spaces,” to appear in
*Forum Mathematicum*. - J.-O. Strömberg and A. Torchinsky,
*Weighted Hardy Spaces*, vol. 1381 of*Lecture Notes in Mathematics*, Springer, Berlin, Germany, 1989. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - A. Jonsson, “Besov spaces on closed subsets of ${\mathbb{R}}^{n}$,”
*Transactions of the American Mathematical Society*, vol. 341, no. 1, pp. 355–370, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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
- 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - H. Federer,
*Geometric Measure Theory*, vol. 153 of*Die Grundlehren der mathematischen Wissenschaften*, Springer, New York, NY, USA, 1969. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Th. Varopoulos, “Analysis on Lie groups,”
*Journal of Functional Analysis*, vol. 76, no. 2, pp. 346–410, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - J. W. Jenkins, “Growth of connected locally compact groups,”
*Journal of Functional Analysis*, vol. 12, pp. 113–127, 1973. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - K. Yosida,
*Functional Analysis*, Classics in Mathematics, Springer, Berlin, Germany, 1995. View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - Y. Meyer, “Les nouveaux opérateurs de Calderón-Zygmund,”
*Astérisque*, no. 131, pp. 237–254, 1985. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. Meyer and R. Coifman,
*Wavelets*, vol. 48 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1997. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - D. Goldberg, “A local version of real Hardy spaces,”
*Duke Mathematical Journal*, vol. 46, no. 1, pp. 27–42, 1979. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Da. Yang, Do. Yang, and Y. Zhou, “Localized Campanato spaces on RD-spaces and their applications to Schrödinger operators,” submitted.
- 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 · View at MathSciNet - Y. Meyer,
*Wavelets and Operators*, vol. 37 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1992. View at Zentralblatt MATH · View at MathSciNet - C. Fefferman and E. M. Stein, “Some maximal inequalities,”
*American Journal of Mathematics*, vol. 93, no. 1, pp. 107–115, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Duoandikoetxea,
*Fourier Analysis*, vol. 29 of*Graduate Studies in Mathematics*, American Mathematical Society, Providence, RI, USA, 2001. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at MathSciNet - Y. Han and D. Yang, “New characterization of BMO $({\mathbb{R}}^{n})$ space,”
*Boletín de la Sociedad Matemática Mexicanae. Tercera Serie*, vol. 10, no. 1, pp. 95–103, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - J. Bergh and J. Löfström,
*Interpolation Spaces. An Introduction*, Springer, Berlin, Germany, 1976. View at Zentralblatt MATH · View at MathSciNet - H. Triebel,
*Interpolation Theory, Function Spaces, Differential Operators*, Johann Ambrosius Barth, Heidelberg, Germany, 2nd edition, 1995. View at Zentralblatt MATH · View at MathSciNet - 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 · View at Zentralblatt MATH · View at MathSciNet - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet