Paracommutators and Minimal Spaces

Part of the NATO ASI series book series (ASIC, volume 153)


The purposes of these lectures is twofold.


Banach Space Holomorphic Function Besov Space Interpolation Space Carleson Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A1]
    Adams, R.A., “Sobolev spaces”. Academic Press, New York —San Fransisco, 1975.zbMATHGoogle Scholar
  2. [A2]
    Ahlmann, M., “The trace ideal criterion for Hankel operators on the weighted Bergman space Aα2 in the unit ball of Cn”. Technical report, Lund, 1984.Google Scholar
  3. [A3]
    Anderson, J.M., Clunie, J. and Pommerenke, C., “On Bloch functions and normal functions”. J. reine angew. Math. 270 (1974), pp. 12–37.MathSciNetzbMATHGoogle Scholar
  4. [A4]
    Arazy, J. and Fisher, S., “The uinqueness of the Dirichlet space among Möbius-invariant Hilbert spaces”. Technical report, Haifa, 1983.Google Scholar
  5. [A5]
    Arazy, J. and Fisher, S., “Some aspects of the minimal, Möbius invariant space of analytic functions on the unit disc”. In: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 24–44. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 1984.CrossRefGoogle Scholar
  6. [A6]
    Aronszajn, N. and Gagliardo, E., “Interpolation spaces and interpolation methods”. Ann. Mat. Pura Appl. 68 (1965), pp. 51–118.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [B1]
    Bergh, J. and Löfström, J., “Interpolation spaces. An introduction”. Grundlehren 223. Springer-Verlag, Berlin — Heidelberg — New York, 1976.zbMATHGoogle Scholar
  8. [B2]
    Besov, O. V., “Investigation of a family of function spaces in connection with theorems of imbedding and extension”. Trudy Mat. Inst. Steklov. 60 (1961), pp. 42–81 С Russian].MathSciNetzbMATHGoogle Scholar
  9. [B3]
    Besov, O.V., Il’in, V.P. and Nikol’skii, S.M., “Integral representations of functions and imbedding theorems, I-II”. John Wiley, New York — Toronto — London — Sydney, 1978–79.Google Scholar
  10. [B4]
    Birman, M.Sh. and Solomyak, M.Z., “Estimates for singular numbers of integral operators.” Uspehi Mat. Nauk 32:1 (1977), pp. 17–84 С Russian].MathSciNetzbMATHGoogle Scholar
  11. [B5]
    Bony, J.-M., “Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles nonlinéaires”. Ann. Sci. Ecole Norm. Sup. 14 (1981), pp, 209–246.MathSciNetzbMATHGoogle Scholar
  12. [B6]
    Brudnyi, Yu. A., “Piece wise polynomial approximation, embedding theorems and rational approximation”. In: Lecture Notes in Mathematics 556, pp. 73–98. Springer-Verlag, Berlin — Heidelberg — New York, 1976.Google Scholar
  13. [B7]
    Brudnyǐ, Yu. A. and Kruglyak, N. Ya., “Real interpolation functors”. Dokl. Akad. Nauk SSSR 250 (1981), pp. 14–17 С Russian].Google Scholar
  14. [B8]
    Brudnyǐ, Yu. A. and Kruglyak, N. Ya., “Real interpolation functors”. Yaroslavl’, 1981 [Russian; English translation in preparation].Google Scholar
  15. [B9]
    Butzer, P.L. and Berens, H., “Semi-groups of operators and approximation”. Grundlehren 145. Springer-Verlag, Berlin — Heidelberg — New York, 1967.zbMATHGoogle Scholar
  16. [C1]
    Calderón, A.P., “Intermediate spaces and interpolation, the complex method.” Studia Math. 24 (1964), pp. 113–190.MathSciNetzbMATHGoogle Scholar
  17. [C2]
    Calderón, A.P. “Spaces between L1 and L and the theorem of Marcinkiewicz”. Studia Math. 26 (1966), pp. 273–299.MathSciNetzbMATHGoogle Scholar
  18. [C3]
    Ceauşu, T. and Gaşpar, T., “A bibliographie on interpolation of operators and applications in comutative and non-comutative harmonic analysis”. [Authors’ spelling!] Seminarul de Operatori Liniari şi Analiză A rmonică, special issue of 1980. Universitatea din Timişoara, Secţia de Matematică.Google Scholar
  19. [C4]
    Cima, J.A., “The basic properties of Bloch functions”. Internat. J. Math. Sci. 2 (1979), pp. 369–413.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [C5]
    Coífman, R. and Rochberg, R., “Representation theorems for holomorphic functions in LP”. Astérisque 77 (1980), pp. 11–66.zbMATHGoogle Scholar
  21. [C6]
    Coífman, R., Rochberg, R. and Weiss, G., “Factorization theorems for Hardy spaces in several variables”. Ann. Math. 103 (1976), pp. 611–635.zbMATHCrossRefGoogle Scholar
  22. [C7]
    Coífman, R., Cwikel, M., Rochberg, R., Sagher, Y. and Weiss, G., “A theory of complex interpolation for families of Banach spaces”. Advances Math. 43 (1982), pp. 202–229.CrossRefGoogle Scholar
  23. [C8]
    Cwikel, M., “K-divisibility of the K-functionl and Calderon pairs”. Ark. Mat. 22 (1964), pp. 39–62.MathSciNetCrossRefGoogle Scholar
  24. [C9]
    Cwikel, M. and Nilsson, P., “The coincidence of the real and the complex interpolation methods for couples of weighted Banach lattices”. In: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 54–65. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 1984.CrossRefGoogle Scholar
  25. [D]
    Dmitriev, V.I., Krein, S.G. and Ovcinnikov, V.I., “Fundamentals of the theory of interpolation of linear operators”. In: Geometry of linear spaces and operator theory, pp. 31–74. Yaroslavl’, 1977 [Russian].Google Scholar
  26. [F1]
    Fefferman, C., Riviere, N. and Sagher, Y., “Interpolation between Hp spaces”. Trans. Amer. Math. Soc. 191 (1974), pp. 75–81.MathSciNetzbMATHGoogle Scholar
  27. [F2]
    Fefferman, C. and Stein, E., “Hp spaces of several variables”. Acta Math. 129 (1972), pp. 137–193.MathSciNetzbMATHCrossRefGoogle Scholar
  28. [F3]
    Flett, T.M., “Lipschitz spaces of functions on the circle and the disc”. J. Math. Anal. Appl. 39 (1972), pp, 125–158.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [F4]
    Foias, C. and Lions, J.-L., “Sur certains espaces d’interpolation”. Acta Szeged 22 (1961), pp. 262–282.MathSciNetGoogle Scholar
  30. [F5]
    Frazier, M. and Jawerth, B., “Decomposition of Besov spaces”. Preprint, 1984.Google Scholar
  31. [G1]
    Gagliardo, E., “Proprietà di alcune classi di funzioni in più variabili”. Ricerche Mat. 7 (1958), pp. 102–137.MathSciNetzbMATHGoogle Scholar
  32. [G2]
    Gagliardo, E., “A common structure in various families of functional spaces. Part II. Ricerche Mat. 12 (1963), pp. 87–107.MathSciNetzbMATHGoogle Scholar
  33. [G3]
    Gagliardo, E., “Caratterizzazione construtiva di tutti gli spazi di interpolazione tra spazi di Banach”. Symposia Mathematica 2 (1968), pp. 95–106.Google Scholar
  34. [G4]
    Grisvard, P., “Commutativité de deux foncteurs d’interpolation et applications”. J. Math. Pures Appl. 45 (1966), pp. 143–290.MathSciNetzbMATHGoogle Scholar
  35. [G5]
    Gustavsson, J. and Peetre, J., “Interpolation of orlicz spaces”. Studia Math. Studia Math. 60 (1977), pp. 33–59.MathSciNetzbMATHGoogle Scholar
  36. [H1]
    Hahn, K.T. and Mitchell, J., “Representation of linear functional s of Hp spaces over bounded symmetric domains in Cn”. J. Math. Anal. Appl. 56 (1976), pp. 379–391.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [H2]
    Hardy, G.H., “Collected papers, III”. Clarendon, Oxford, 1969.Google Scholar
  38. [H3]
    Hörmander, L., “Linear differential operators”. Grundlehren 116. Springer-Verlag, Berlin — Göttingen — Heidelberg, 1961.Google Scholar
  39. [H4]
    Horowicz, C. and Oberlin, D., “Restrictions of Hp functions to the diagonal of Un”. Indiana U. Math. J. 24 (1975), pp. 767–772.CrossRefGoogle Scholar
  40. [H5]
    Hastings, W. H., “A Carleson measure theorem for Bergman spaces”. Proc. Amer. Math. Soc. 52 (1975), pp. 237–241.MathSciNetzbMATHCrossRefGoogle Scholar
  41. [I]
    Irodova, I.P., “On the properties of the scale of spaces Bλθ for 0 < p < 1”. Dokl. Akad. Nauk SSSR 250 (1980), pp. 273–275 [Russian]p.MathSciNetGoogle Scholar
  42. [J1]
    Janson, S., “Mean oscillation and commutators of singular integral operators”. Ark. Mat. 16 (1978), pp. 263–270.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [J2]
    Janson, S., “Generalizations of Lipschiytz spaces and application to Hardy spaces and bounded mean oscillation”. Duke J. Math. 27 (1980), pp. 959–982.MathSciNetCrossRefGoogle Scholar
  44. [J3]
    Janson, S., “Minimal and maximal methods of interpolation”. J. Funct. Anal. 44 (1981), pp. 50–73.MathSciNetzbMATHCrossRefGoogle Scholar
  45. [J4]
    Janson, S., Nilsson, P. and Peetre, J., “Notes on Wolff’s note on interpolation spaces”. Proc. London Math. Soc. 48 (1984), pp. 283–299.MathSciNetzbMATHCrossRefGoogle Scholar
  46. [J5]
    Janson, S. and Peetre, J., “Higher order commutators of singular integral operators”. In: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 125–142. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 1984.CrossRefGoogle Scholar
  47. [J6]
    Janson, S., Peetre, J. and Semmes, S., “On the action of Hankel and Toeplitz operators on some function spaces”. Technical report, Uppsala, 1984 [to appear in Duke Math. J.3.Google Scholar
  48. [J7]
    Janson, S. and Wolff, T., “Schatten classes and commutators of singular integral operators”. Ark. Mat. 20 (1982), pp. 301–310.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [J8]
    Jonsson, A. and Wallin, H., “Function spaces on subsets of Rn”. Math. Reports 2 (1984), pp, 1–221.MathSciNetGoogle Scholar
  50. [J9]
    Jordan, C., “Réduction d’un réseau de formes quadratiques ou bilinéaires”. J. Math. Pures Appl. 2 (1906), pp. 403–438, 3 (1907), pp. 5–51 [reprinted in Oeuvres, t. III, pp. 269–350. Gauthiers-Villars, Paris, 19623.Google Scholar
  51. [K1]
    Krantz, S., “Intrinsic Lipschitz classes on manifolds with applications to complex function theory and estimates for the ≙̄ and ≙̄b equations”. Manuscripta Math. 24 (1978), pp. 351–378.MathSciNetzbMATHCrossRefGoogle Scholar
  52. [K2]
    Kreǐn, S.G., Petunin, Yu.I. and Semenov, E.M., “Interpolation of linear operators”. Izd. Nauka, Moscow, 1978 [Russian; English translation: A.M.S., Providence, 19823.Google Scholar
  53. [L1]
    Lions, J.-L., “Espaces intermédiaires entre espaces hilbertiens et applications”. Bull. Math. Soc. Sci. Math. Phys. R.P. Roumanie 50 (1958), pp. 419–432.Google Scholar
  54. [L2a]
    Lions, J.-L., “Théoremes de traces et d’interpolation, I–V”. Ann. Scuola Norm. Sup. Pisa 13 (1959), pp. 389–403, 14 (1960), pp. 317–331;MathSciNetGoogle Scholar
  55. [L2b]
    Lions, J.-L., “Théoremes de traces et d’interpolation, I–V”. J. Math. Pures Appl. 42 (1963), pp. 195–203;MathSciNetzbMATHGoogle Scholar
  56. [L2c]
    Lions, J.-L., “Théoremes de traces et d’interpolation, I–V”. Math. Ann. 151 (1963), pp. 42–56;MathSciNetzbMATHCrossRefGoogle Scholar
  57. [L2d]
    Lions, J.-L., “Théoremes de traces et d’interpolation, I–V”. An. Acad. Brasiliera Cien. 35 (1963), pp. 1–10.Google Scholar
  58. [L3]
    Lions, J.-L., “Equations différentielles opérationnelles et problèmes mixtes”. Grundlehren 111. Springer-Verlag, Berlin — Göttingen — Heidelberg, 1961.Google Scholar
  59. [L4]
    Lions, J.-L. and Magenes, E., “Problemes aux limites non homogènes et applications, I”. Dunod, Paris, 1968.Google Scholar
  60. [L5]
    Lions, J.-L. and Peetre, J., “Sur une classes d’espaces d’interpolation”. Publ. Math. Inst. Hautes Etudes Sci. 19 (1964), pp. 5–68.MathSciNetzbMATHCrossRefGoogle Scholar
  61. [M]
    Mityagin, B., “An interpolation theorem for modular spaces”. Mat Sbornik 66 (1965), pp. 473–482 [Russian; English translation in: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 10–23. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 19843.Google Scholar
  62. [N1]
    Nikol’skii, S.M., “Approximation of functions of several variables and imbedding theorems”. Grundlehren 205. Springer-Verlag, Berlin — Heidelberg — New York, 1975.Google Scholar
  63. [N2]
    Nilsson, P., “Reiteration theorems for real interpolation and approximation spaces”. Ann. Mat. Pura Appl. 132 (1982), pp. 291–330.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [N3]
    Nilsson, P., “Interpolation of Calderon pairs”. Ann. Mat. Pura Appl. 134 (1983), pp. 201–232.MathSciNetzbMATHCrossRefGoogle Scholar
  65. [O1]
    Oklander, E.T., “Λpq interpolators and the theorem of Marcinkiewicz”. Bull. Amer. Math.Soc. 72 (1966), pp. 49–53.MathSciNetzbMATHCrossRefGoogle Scholar
  66. [O2]
    Ovchinnikov, V.I., “Interpolation theorems resulting from Grothendieck’s inequality”. Funkcional. Anal. i Prilozen. 10 (1976), pp. 45–54.zbMATHCrossRefGoogle Scholar
  67. [O3]
    Ovčinnikov, V. I., “The method of orbits in interpolation theory”. Math. Reports 1 (1984), pp. 349–515.Google Scholar
  68. [P1]
    Peetre, J., “Espaces d’interpolation et théoreme de Soboleff”. Ann. Inst. Fourier 16 (1966), pp. 279–317.MathSciNetzbMATHCrossRefGoogle Scholar
  69. [P2]
    Peetre, J., “New thoughts on Besov spaces”. Duke Univ. Math. Series 1. Durham, 1976.zbMATHGoogle Scholar
  70. [P3]
    Peetre, J., “Invariant function spaces connected with the holomorphic discrete series”. In: Butzer, P.L. et al (eds.), “Anniversary volume on Approximation Theory and Functional Analysis”, pp. 119–134. Birkhäuser, Basel, Boston, 1984. [Also available in the Lund technical report series.]Google Scholar
  71. [P4]
    Peetre, J., “Recent progress in real interpolation”. In: “Methods of Functional Analysis and Theory of Elliptic Equations Proceedings of the International Meeting dedicated to the memory of professor Carlo Miranda, Naples, September 13–16, 1982”, pp. 231–263. Liguori, Naples, 1983.Google Scholar
  72. [P5]
    Peetre, J., “The theory of interpolation — its origin, prospects for the future”. In: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 1–9. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 1984.CrossRefGoogle Scholar
  73. [P6]
    Peller, V.V., “Hankel operators of class Sp and applications (rational approximation, Gaussian processes, the majorant problem for operators)”. Mat. Sb. 113 (1980), pp. 538–581 [Russian].MathSciNetGoogle Scholar
  74. [P7]
    Peller, V.V., “Hankel operators of the Schatten-von Neumann class Sp, 0 < p < 1.” LOMI preprints E-6–82, Leningrad, 1982.Google Scholar
  75. [P8]
    Peller, V.V., “Estimates of functions of power bounded operators in Hilbert space”. J. Operator Theory 7 (1982), pp, 341–372.MathSciNetzbMATHGoogle Scholar
  76. [P9]
    Persson, L.-E., “Description of some interpolation spaces in off-diagonal cases”. In: “Interpolation Spaces and Allied Topics in Analysis. Proceedings, Lund, 1983”. Lecture Notes in Mathematics 1070, pp. 213–231. Springer-Verlag, Berlin — Heidelberg — New York — Tokyo, 1984.CrossRefGoogle Scholar
  77. [P10]
    Pietsch, A., “Ideals of multilinear functionals”. Preprint, Jena, 1983.Google Scholar
  78. [R1]
    Rivière, N.M. and Sagher, Y., “Interpolation between L and H1, the real method”. J. Funct. Anal. 14 (1973), pp. 401–409.zbMATHCrossRefGoogle Scholar
  79. [R2]
    Rochberg, R., “Trace ideal criteria for Hankel operators and commutators”. Indiana Univ. Math. J. 31 (1982), pp. 913–925.MathSciNetzbMATHCrossRefGoogle Scholar
  80. [R3]
    Russo, A., “On the Hausdorff-Young theorem for integral operators”. Pac. J. Math. 16 (1977), pp. 241–253.Google Scholar
  81. [R4]
    Rubel, L.A. and Timoney, R.M., “An extremal property of the Bloch space”. Proc. Amer. Math. Soc. 75 (1979), pp. 45–49.MathSciNetzbMATHCrossRefGoogle Scholar
  82. [R5]
    Rochberg, R. and Weiss, G., “Derivatives of analytic families of Banach spaces”. Ann. Math. 118 (1983), pp. 315–347.MathSciNetzbMATHCrossRefGoogle Scholar
  83. [S1]
    Semmes, S., “Trace ideal criterion for Hankel operators, 0 < p < 1”. Integral Equations Operator Theory 7 (1984), pp. 241–281.MathSciNetzbMATHCrossRefGoogle Scholar
  84. [S2]
    Shapiro, H.S., “A Tauberian theorem related to approximation theory”. Acta Math. 120 (1968), pp. 279–292.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [S3]
    Shields, A., “The analogue of the Fejér-Riesz theorem for the Dirichlet space”. In: Beckner, W. et al (eds.), “Conference in Harmonic Analysis in honor of Antoni Zygmund”, vol. II, pp. 810–820. Wadsworth, Belmont, 1983.Google Scholar
  86. [S4]
    Stein, E.M., “Singular integrals and differentiability properties of functions”. Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar
  87. [S5]
    Stein, E.M., “Singular integrals and estimates for the Cauchy-Riemann equations”. Bull. Amer. Math. Soc. 79 (1973), pp. 440–445.MathSciNetzbMATHCrossRefGoogle Scholar
  88. [S6]
    Stein, E.M. and Weiss, G., “Interpolation of operators with change of measure”. Trans. Amer. Math. Soc. 87 (1958), pp. 159–172.MathSciNetzbMATHCrossRefGoogle Scholar
  89. [S7]
    Strichartz, R., “Para-differential operators — another step forward for the method of Fourier”. Notices Amer. Math. Soc. 29 (1982), pp. 440–445.Google Scholar
  90. [T1]
    Taibleson, M., “On the theory of Lipschitz spaces of distributions of Euclidean n-space. I–II”. J. Math. Mech. 13 (1964), pp. 821–839.MathSciNetGoogle Scholar
  91. [T2]
    Triebel, H., “Interpolation theory. Functions spaces. Differential operators”. VEB, Berlin, 1978.Google Scholar
  92. [T3]
    Triebel, H., “Theory of function spaces”. Akademische Verlagsgesellschaft Geest & Portig, Leipzig, 1983.CrossRefGoogle Scholar
  93. [T4]
    Timotin, D., “A note on Cp estimates for certain kernels”. Preprint, Bucuresti, 1984.Google Scholar
  94. [U]
    Uchiyama, A., “On the compactness of operators of Hankel type”. Tohoku Math. J. 30 (1978), pp. 163–171.MathSciNetzbMATHCrossRefGoogle Scholar
  95. [W]
    Wolff, T., “A note on interpolation spaces”. In: “Proc. Conf. Harmonic Analysis, Univ. of Minnesota, Minneapolis”. Lecture Notes in Mathematics 908, pp. 199–204. Springer-Verlag, Berlin — Heidelberg — New York, 1982.Google Scholar
  96. [Z]
    Zygmund, A., “Trigonometrical series I–II”. Cambridge University Press, Cambridge, 1958.Google Scholar

Copyright information

© D. Reidel Publishing Company 1985

Authors and Affiliations

  1. 1.University of LundLundSweden

Personalised recommendations