• Dorina Mitrea
  • Irina Mitrea
  • Marius Mitrea
  • Sylvie Monniaux
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


In this chapter we state a general metrization theorem, in the algebraic setting of groupoids, and explain its relationship to several classical results in analysis such as the Macías–Segovia metrization theorem for quasimetric spaces, the Aoki–Rolewicz theorem for quasinormed vector spaces, and the Alexandroff–Urysohn metrization theorem for uniform spaces. The metrization theorem in question is quantitative in nature and involves starting from a given quasisubadditive function defined on the underlying groupoid. We also indicate that our general metrization theorem is sharp.


Hardy Space Topological Vector Space Uniform Space Homogeneous Type Normed Vector Space 



In the early stages of its inception, various parts of this monograph were used to teach several topic courses at the graduate level at the University of Missouri. We wish to take this opportunity to thank our students, especially Ryan Alvarado, Kevin Brewster, Dan Brigham, Brock Schmutzler, and Elia Ziadé, for their active participation and their careful reading of preliminary notes. The authors also gratefully acknowledge the support of the Simons Foundation Grant No. 200750 as well as the US NSF Grants DMS-1201736 and DMS-0653180. Last but not least, the authors thank the anonymous referees for making a number of useful suggestions, which have improved the presentation.


  1. 1.
    L.V. Ahlfors, Bounded analytic functions. Duke Math. J. 14, 1–11 (1947)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    H. Aimar, B. Iaffei, L. Nitti, On the Macías-Segovia metrization theorem of quasi-metric spaces. Revista U. Mat. Argentina 41, 67–75 (1998)MathSciNetMATHGoogle Scholar
  3. 3.
    F. Albiac, N.J. Kalton, Lipschitz structure of quasi-Banach spaces. Israel J. Math. 170, 317–335 (2009)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    P. Alexandroff, P. Urysohn, Une condition nécessaire et suffisante pour qu’une classe (L) soit une classe (D). C. R. Acad. Sci. Paris 177, 1274–1277 (1923)Google Scholar
  5. 5.
    R. Alvarado, D. Mitrea, I. Mitrea, M. Mitrea, Weighted mixed-normed spaces on quasi-metric spaces, preprint (2012)Google Scholar
  6. 6.
    I. Amemiya, A generalization of Riesz-Fischer’s theorem. J. Math. Soc. Jpn. 5, 353–354 (1953)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    T. Aoki, Locally bounded topological spaces. Proc. Jpn. Acad. Tokyo 18, 588–594 (1942)MATHCrossRefGoogle Scholar
  8. 8.
    N. Aronszajn, Quelques remarques sur les relations entre les notions d’écart régulier et de distance. Bull. Am. Math. Soc. 44, 653–657 (1938)MathSciNetCrossRefGoogle Scholar
  9. 9.
    P. Assouad, Espaces métriques, plongements, facteurs. Thèse de doctorat d’État, Orsay, 1977MATHGoogle Scholar
  10. 10.
    P. Assouad, Étude d’une dimension métrique liée à la possibilité de plongements dans \({\mathbb{R}}^{n}\). C. R. Acad. Sci. Paris, Série A 288, 731–734 (1979)Google Scholar
  11. 11.
    P. Assouad, Plongements Lipschitziens dans \({\mathbb{R}}^{n}\). Bull. Soc. Math. France 111, 429–448 (1983)MathSciNetMATHGoogle Scholar
  12. 12.
    S. Banach, Metrische Gruppen. Studia Math. 3, 101–113 (1931)Google Scholar
  13. 13.
    S. Banach, Théorie des Opérations Linéaires, Warsaw, 1932Google Scholar
  14. 14.
    A. Benedek, R. Panzone, The space L P, with mixed norm. Duke Math. J. 28(3), 301–324 (1961)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    C. Bennett, R. Sharpley, Interpolation of operators. Pure and Applied Mathematics, vol. 129 (Academic, New York, 1988)Google Scholar
  16. 16.
    J. Bergh, J. Löfström, Interpolation Spaces. An Introduction (Springer, Berlin, 1976)Google Scholar
  17. 17.
    A.S. Besicovitch, I.J. Schoenberg, On Jordan arcs and Lipschitz classes of functions defined on them. Acta Math. 106, 113–136 (1961)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    R.H. Bing, Metrization of topological spaces. Can. J. Math. 3, 175–186 (1951)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    G. Birkhoff, A note on topological groups. Compositio Math. 3, 427–430 (1956)MathSciNetGoogle Scholar
  20. 20.
    N. Bourbaki, Topologie générale, Chapitre 9. Utilisation des nombres réels en topologie générale (Act. Sci. Ind. 1045) (Hermann, Paris, 1958)Google Scholar
  21. 21.
    H. Brandt, Über eine Verallgemeinerung des Gruppenbegriffes. Math. Annalen 96, 360–366 (1926)CrossRefGoogle Scholar
  22. 22.
    L.G. Brown, Note on the open mapping theorem. Pac. J. Math. 38(1), 25–28 (1971)MATHCrossRefGoogle Scholar
  23. 23.
    R. Brown, From groups to groupoids: a brief survey. Bull. Lond. Math. Soc. 19, 113–134 (1987)MATHCrossRefGoogle Scholar
  24. 24.
    R. Brown, Topology and Groupoids (BookSurge Publishing, 2006)Google Scholar
  25. 25.
    R.H. Bruck, A Survey of Binary Systems (Ergebnisse der Mathematik N.F. 20) (Springer, Berlin, 1958)Google Scholar
  26. 26.
    Y. Brudnyĭ, N. Krugljak, Interpolation Functors and Interpolation Spaces, vol. I (North-Holland, Amsterdam, 1991)MATHGoogle Scholar
  27. 27.
    D. Burago, Y. Burago, S.V. Ivanov, A Course in Metric Geometry (American Mathematical Society, Providence, 2001)MATHGoogle Scholar
  28. 28.
    F. Cabello Sánchez, J.M.F. Castillo, Banach space techniques underpinning a theory for nearly additive mappings, Dissertationes Math. (Rozprawy Mat.), vol. 404, 2002Google Scholar
  29. 29.
    J. Cerdà, J. Martín, P. Silvestre, Capacitary function spaces. Collect. Math. 62(1), 95–118 (2011)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    E.W. Chittenden, On the equivalence of écart and voisinage. Trans. Am. Math. Soc. 18, 161–166 (1917)MathSciNetMATHGoogle Scholar
  31. 31.
    E.W. Chittenden, On the metrization problem and related problems in the theory of abstract sets. Bull. Am. Math. Soc. 33, 13–34 (1927)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    M. Christ, in Lectures on Singular Integral Operators. CBMS Regional Conference Series in Mathematics, vol. 77 (American Mathematical Society, Providence, 1990)Google Scholar
  33. 33.
    R.R. Coifman, Y. Meyer, E.M. Stein, Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335 (1985)MathSciNetMATHCrossRefGoogle Scholar
  34. 34.
    R.R. Coifman, G. Weiss, in Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. Lecture Notes in Mathematics, vol. 242 (Springer, Berlin, 1971)Google Scholar
  35. 35.
    R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4) 569–645 (1977)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    J. Cygan, Subadditivity of homogeneous norms on certain nilpotent Lie groups. Proc. Am. Math. Soc. 83, 69–70 (1981)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    G. David, J.L. Journé, S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Math. Iberoam. 1, 1–56 (1985)MATHCrossRefGoogle Scholar
  38. 38.
    G. David, S. Semmes, in Fractured Fractals and Broken Dreams: Self-similar Geometry Through Metric and Measure. Oxford Lecture Series in Mathematics and its Applications, vol. 7 (Clarendon, Oxford University Press, New York, 1997)Google Scholar
  39. 39.
    D. Deng, Y. Han, in Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics, vol. 1966 (Springer, Berlin, 2009)Google Scholar
  40. 40.
    A. Di Concilio, S.A. Naimpally, A unified approach to metrization problems. Acta Math. Hungarica 53(1–2), 109–113 (1998)Google Scholar
  41. 41.
    J. Dieudonné, L. Schwartz, La dualité dans les espaces (F) et (LF). Ann. Inst. Fourier, Grenoble 1 (1949), 61–101 (1950)Google Scholar
  42. 42.
    J.J. Dudziak, Vitushkin’s Conjecture for Removable Sets (Universitext) (Springer, Berlin, 2010)Google Scholar
  43. 43.
    V.A. Efremovič, A.S. Švarc, A new definition of uniform spaces. Metrization of proximity spaces, (Russian) Doklady Akad. Nauk SSSR (N.S.) 89, 393–396 (1953)Google Scholar
  44. 44.
    R. Engelking, General Topology (Heldermann, Berlin, 1989)MATHGoogle Scholar
  45. 45.
    G.B. Folland, E. Stein, Hardy Spaces on Homogeneous Groups (Princeton University Press, Princeton, 1982)MATHGoogle Scholar
  46. 46.
    M. Frazier, B. Jawerth, Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    M. Fréchet, Les dimensions d’une ensemble abstrait. Math. Ann. 68, 145–168 (1909–1910)Google Scholar
  49. 49.
    A.H. Frink, Distance functions and the metrization problem. Bull. Am. Math. Soc. 43, 133–142 (1937)MathSciNetCrossRefGoogle Scholar
  50. 50.
    I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, in Weighted Theory for Integral Transforms on Spaces of Homogeneous Type. Pitman Monographs and Surveys in Pure and Applied Mathematics, Addison Wesley Longman Inc. vol. 92 (1998)Google Scholar
  51. 51.
    A. Gogatishvili, P. Koskela, N. Shanmugalingam, in Interpolation Properties of Besov Spaces Defined on Metric Spaces. Mathematische Nachrichten, Special Issue: Erhard Schmidt Memorial Issue, Part II, vol. 283, Issue 2 (2010), pp. 215–231Google Scholar
  52. 52.
    A. Grothendieck, in Produits Tensoriels Topologique et Espaces Nucléaires. Memoirs of the American Mathematical Society, vol. 16 (AMS, Providence, 1955)Google Scholar
  53. 53.
    J. Gustavsson, Metrization of quasi-metric spaces. Math. Scand. 35, 56–60 (1974)MathSciNetMATHGoogle Scholar
  54. 54.
    P. Hajłasz, Whitney’s example by way of Assouad’s embedding. Proc. Am. Math. Soc. 131(11), 3463–3467 (2003)MATHCrossRefGoogle Scholar
  55. 55.
    Y. Han, D. Müller, D. Yang, A theory of Besov and Triebel-Lizorkin spaces on metric measure spaces modeled on Carnot-Carathéodory spaces. Abstr. Appl. Anal. no. 893409, 1–250 (2008)CrossRefGoogle Scholar
  56. 56.
    Y. Han, E. Sawyer, in Littlewood-Paley Theory on Spaces of Homogeneous Type and the Classical Function Spaces. Memoirs of the American Mathematical Society, vol. 530 (AMS, Providence, 1994)Google Scholar
  57. 57.
    F. Hausdorff, Grundzüge der Mengenlehre (Von Veit, Leipzig, 1914)MATHGoogle Scholar
  58. 58.
    W. Hebisch, A. Sikora, A smooth subadditive homogeneous norm on a homogeneous group. Studia Math. 96(3), 231–236 (1990)MathSciNetMATHGoogle Scholar
  59. 59.
    J. Heinonen, Lectures on Analysis on Metric Spaces, Universitext (Springer, New York, 2001)Google Scholar
  60. 60.
    T. Holmstedt, Interpolation of quasi-normed spaces. Math. Scand. 26, 177–199 (1970)MathSciNetMATHGoogle Scholar
  61. 61.
    L. Hörmander, The Analysis of Linear Partial Differential Operators, vol. I (reprint of the 2-nd edition 1990) (Springer, Berlin, 2003)Google Scholar
  62. 62.
    G. Hu, D. Yang, Y. Zhou, Boundedness of singular integrals in Hardy spaces on spaces of homogeneous type. Taiwanese J. Math. 133(1), 91–135 (2009)MathSciNetGoogle Scholar
  63. 63.
    T. Husain, S-spaces and the open mapping theorem. Pac. J. Math. 12(1), 253–271 (1962)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    T. Husain, Introduction to Topological Groups (W.B. Saunders, Philadelphia, 1966)MATHGoogle Scholar
  65. 65.
    S. Kakutani, Über die Metrisation der topologischen Gruppen. Proc. Imp. Acad. Jpn. 12, 82–84 (1936)MathSciNetCrossRefGoogle Scholar
  66. 66.
    N.J. Kalton, Basic sequences in F-spaces and their applications. Proc. Edinb. Math. Soc. (2) 19(2), 151–167 (1974/1975)Google Scholar
  67. 67.
    N.J. Kalton, in Quasi-Banach spaces, ed. by W.B. Johnson, J. Lindenstrauss. Handbook of the Geometry of Banach Spaces. Chapter 25 in vol. 2 Elsevier Science B. V. (2003)Google Scholar
  68. 68.
    N. Kalton, S. Mayboroda, M. Mitrea, in Interpolation of Hardy-Sobolev-Besov-Triebel-Lizorkin Spaces and Applications to Problems in Partial Differential Equations, ed. by L. De Carli, M. Milman. Interpolation Theory and Applications. Contemporary Mathematics, vol. 445 (American Mathematical Society, Providence, 2007), pp. 121–177Google Scholar
  69. 69.
    N.J. Kalton, N.T. Peck, J.W. Roberts, in An F-space Sampler. London Mathematical Society Lecture Notes Series, vol. 89 (Cambridge University Press, Cambridge, 1984)Google Scholar
  70. 70.
    A. Kamińska, Some remarks on Orlicz-Lorentz spaces. Math. Nachr. 147, 29–38 (1990)MathSciNetMATHCrossRefGoogle Scholar
  71. 71.
    J.L. Kelley, General Topology (van Nostrand, Toronto, 1955)Google Scholar
  72. 72.
    M.D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22, 77–108 (1934)Google Scholar
  73. 73.
    P. Koskela, N. Shanmugalingam, H. Tuominen, Removable sets for the Poincaré inequality on metric spaces. Indiana Math. J. 49, 333–352 (2000)MathSciNetMATHCrossRefGoogle Scholar
  74. 74.
    G. Köthe, Topological Vector Spaces I (Springer, Berlin, 1969)MATHCrossRefGoogle Scholar
  75. 75.
    C. Kuratowski, Quelques problèmes concernant les espaces métriques non-séparables. Fund. Math. 25, 534–545 (1935)Google Scholar
  76. 76.
    S. Leader, Metrization of proximity spaces. Proc. Am. Math. Soc. 18, 1084–1088 (1967)MathSciNetMATHCrossRefGoogle Scholar
  77. 77.
    J. Luukkainen, H. Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces. Fund. Math. 144, 181–193 (1994)MathSciNetMATHGoogle Scholar
  78. 78.
    J. Luukkainen, E. Saksman, Every complete doubling metric space carries a doubling measure. Proc. Am. Math. Soc. 126(2), 531–534 (1998)MathSciNetMATHCrossRefGoogle Scholar
  79. 79.
    R.A. Macías, C. Segovia, Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270 (1979)MATHCrossRefGoogle Scholar
  80. 80.
    R.A. Macías, C. Segovia, A decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math. 33(3), 271–309 (1979)MATHCrossRefGoogle Scholar
  81. 81.
    E.J. McShane, Extension of range of functions. Bull. Am. Math. Soc. 40, 837–842 (1934)MathSciNetCrossRefGoogle Scholar
  82. 82.
    D. Mitrea, I. Mitrea, M. Mitrea, E. Ziadé, Abstract capacitary estimates and the completeness and separability of certain classes of non-locally convex topological vector spaces. J. Funct. Anal. 262, 4766–4830 (2012)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    I. Mitrea, M. Mitrea, E. Ziadé, in A quantitative Open Mapping Theorem for quasi-pseudonormed groups, Advances in Harmonic Analysis and Applications, a volume in honor of K.I. Oskolkov, Springer Proceedings in Mathematics, 25, 259–286 (2013)Google Scholar
  84. 84.
    D. Montgomery, L. Zippin, Topological Transformation Groups (Interscience Publishers, New York, 1955)MATHGoogle Scholar
  85. 85.
    S. Montgomery-Smith, in Boyd indices of Orlicz-Lorentz spaces. Function Spaces (Edwardsville, IL, 1994). Lecture Notes in Pure and Applied Mathematics, vol. 172 (Dekker, New York, 1995), pp. 321–334Google Scholar
  86. 86.
    P.S. Muhly, Coordinates in Operator Algebras, book manuscript (1997)Google Scholar
  87. 87.
    J.R. Munkres, Topology, 2nd edn. (Prentice Hall, Englewood Cliffs, NJ, 2000)MATHGoogle Scholar
  88. 88.
    J. Nagata, On a necessary and sufficient condition of metrizability. J. Inst. Polytech. Osaka City Univ. Ser. A. Math. 1, 93–100 (1950)MathSciNetGoogle Scholar
  89. 89.
    F. Nazarov, S. Treil, A. Vol’berg, Tb-theorem on non-homogeneous spaces. Acta Math. 190(2), 151–239 (2003)Google Scholar
  90. 90.
    V.W. Niemytzki, On the third axiom of metric spaces. Trans. Am. Math. Soc. 29, 507–513 (1927)MathSciNetMATHGoogle Scholar
  91. 91.
    S. Okada, W.J. Ricker, E.A. Sánchez Pérez, in Optimal Domain and Integral Extension of Operators. Operator Theory, Advances and Applications, vol. 180 (Birkhäuser, Basel, 2008)Google Scholar
  92. 92.
    J.C. Oxtoby, Cartesian products of Baire spaces. Fund. Math. 49, 157–166 (1961)MathSciNetMATHGoogle Scholar
  93. 93.
    M. Paluszyński, K. Stempak, On quasi-metric and metric spaces. Proc. Am. Math. Soc. 137, 4307–4312 (2009)MATHCrossRefGoogle Scholar
  94. 94.
    P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. 129, 1–60 (1989)MathSciNetMATHCrossRefGoogle Scholar
  95. 95.
    A.R. Pears, Dimension Theory of General Spaces (Cambridge University Press, London, 1975)MATHGoogle Scholar
  96. 96.
    J. Peetre, Espaces d’interpolation, généralisations, applications. Rend. Sem. Mat. Fis. Milano 34, 133–164 (1964)MathSciNetMATHCrossRefGoogle Scholar
  97. 97.
    J. Peetre, G. Sparr, Interpolation of normed Abelian groups. Ann. Math. Pura Appl. 92, 217–262 (1972)MathSciNetMATHCrossRefGoogle Scholar
  98. 98.
    B. Pettis, On continuity and openness of homomorphisms in topological groups. Ann. Math. 54, 293–308 (1950)MathSciNetCrossRefGoogle Scholar
  99. 99.
    A. Pietsch, History of Banach Spaces and Linear Operators (Birkhäuser, Boston, 2007)MATHGoogle Scholar
  100. 100.
    D. Pompeiu, Sur la continuité des fonctions de variables complexes (Thèse), Gauthier-Villars, Paris, 1905; Ann. Fac. Sci. de Toulouse 7, 264–315 (1905)MathSciNetMATHGoogle Scholar
  101. 101.
    J. Renault, in A Groupoid Approach to C -Algebras. Lecture Notes in Mathematics, vol. 793 (Springer, Berlin, 1980)Google Scholar
  102. 102.
    A.P. Robertson, W. Robertson, On the closed graph theorem. Proc. Glasgow Math. Ass. 3, 9–12 (1956)MATHCrossRefGoogle Scholar
  103. 103.
    S. Rolewicz, On a certain class of linear metric spaces. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 5, 471–473 (1957)MathSciNetMATHGoogle Scholar
  104. 104.
    S. Rolewicz, Metric Linear Spaces (D. Reidel, Dordrecht, 1985)MATHGoogle Scholar
  105. 105.
    D. Rolfsen, Alternative metrization proofs. Can. J. Math. 18, 750–757 (1966)MathSciNetMATHCrossRefGoogle Scholar
  106. 106.
    H.L. Royden, Real Analysis, 2nd edn. (MacMillan, New York, 1968)Google Scholar
  107. 107.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1976)Google Scholar
  108. 108.
    W. Rudin, in Functional Analysis, 2nd edn. International Series in Pure and Applied Mathematics (McGraw-Hill, New York, 1991)Google Scholar
  109. 109.
    T. Runst, W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Operators (de Gruyter, Berlin, 1996)Google Scholar
  110. 110.
    S. Semmes, Bilipschitz embeddings of metric spaces into Euclidean spaces. Publ. Math. 43(2), 571–653 (1999)MathSciNetMATHGoogle Scholar
  111. 111.
    R. Sikorski, Boolean Algebras (Springer, Berlin, 1960)MATHGoogle Scholar
  112. 112.
    Y. Smirnov, A necessary and sufficient condition for metrizability of a topological space. (Russian) Dokl. Akad. Nauk SSSR (N.S.) 77, 197–200 (1951)Google Scholar
  113. 113.
    E.M. Stein, in Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, No. 30 (Princeton University Press, Princeton, 1970)Google Scholar
  114. 114.
    E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals (Princeton University Press, Princeton, 1993)MATHGoogle Scholar
  115. 115.
    A.H. Stone, Sequences of coverings. Pac. J. Math. 10, 689–691 (1960)MATHGoogle Scholar
  116. 116.
    X. Tolsa, Painlevé’s problem and the semiadditivity of analytic capacity. Acta Math. 190, 105–149 (2003)MathSciNetMATHCrossRefGoogle Scholar
  117. 117.
    X. Tolsa, Analytic capacity, rectifiability, and the Cauchy integral, in Proceedings of the ICM, Madrid, 2006, pp. 1505–1527Google Scholar
  118. 118.
    A. Torchinsky, Interpolation of operators and Orlicz classes. Studia Math. 59, 177–207 (1976)MathSciNetMATHGoogle Scholar
  119. 119.
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, 2nd revised and enlarged edition (Johann Ambrosius Barth, Heidelberg, 1995)Google Scholar
  120. 120.
    H. Triebel, Theory of Function Spaces (Birkhäuser, Berlin, 1983)CrossRefGoogle Scholar
  121. 121.
    H. Triebel, in Theory of Function Spaces, II. Monographs in Mathematics, vol. 84 (Birkhäuser, Basel, 1992)Google Scholar
  122. 122.
    H. Triebel, A new approach to function spaces on spaces of homogeneous type. Rev. Mat. Comput. 18(1), 7–48 (2005)MathSciNetMATHGoogle Scholar
  123. 123.
    H. Triebel, Theory of Function Spaces III (Birkhäuser, Basel, 2006)MATHGoogle Scholar
  124. 124.
    A. Tychonoff, Über einen Metrisationssatz von P. Urysohn. Math. Ann. 95, 139–142 (1926)MathSciNetCrossRefGoogle Scholar
  125. 125.
    P. Urysohn, Zum Metrisationsproblem. Math. Ann. 94, 309–315 (1925)MathSciNetMATHGoogle Scholar
  126. 126.
    D.A. Vladimirov, in Boolean Algebras in Analysis. Mathematics and Its Applications (Kluwer, Dordrecht, 2002)Google Scholar
  127. 127.
    A.L. Vol’berg, S.V. Konyagin, On measures with the doubling condition. Izv. Akad. Nauk SSSR Ser. Mat. 51(3), 666–675 (1987) (Russian); translation in Math. USSR-Izv., 30(3), 629–638 (1988)Google Scholar
  128. 128.
    A. Weil, Sur les espaces à structure uniforme et sur la topologie générale. Act. Sci. Ind. Paris 551 (1937)Google Scholar
  129. 129.
    H. Whitney, Analytic extensions of functions defined on closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)MathSciNetCrossRefGoogle Scholar
  130. 130.
    A.C. Zaanen, Integration (North-Holland, Amsterdam, 1967)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Dorina Mitrea
    • 1
  • Irina Mitrea
    • 2
  • Marius Mitrea
    • 1
  • Sylvie Monniaux
    • 3
  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA
  2. 2.Department of MathematicsTemple UniversityPhiladelphiaUSA
  3. 3.UFR SciencesUniversité Aix-Marseille IIIMarseilleFrance

Personalised recommendations