Methods of the Theory of Singular Integrals: Littlewood-Paley Theory and Its Applications

  • E. M. Dyn’kin
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 42)

Abstract

This article is an immediate continuation to the article “Methods of the Singular Integrals: Hilbert Transform and Calderon-Zygmund Theory”, published in Vol. 15 of this series (Dyn’kin (1987)).

Keywords

Manifold Expense Convolution Hunt Stein 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Aguilera, N. and Segovia, C. (1977): Weighted norm inequalities relating the (math) and the area functions. Stud. Math. 61, No. 3, 293–303, Zbl. 375.28015.MathSciNetMATHGoogle Scholar
  2. Battle, G. and Federbush, P. (1982): A phase cell cluster expansion for Euclidean field theories. Ann. Phys. 142, 95–139.MathSciNetCrossRefGoogle Scholar
  3. Belinskiĭ, P. P. (1974): General properties of quasiconformal mappings. Moscow: Nauka (98 pp.) [Russian] Zbl. 292.30019.Google Scholar
  4. Brackx, F., Delange, R., and Sommer, F. (1982): Clifford analysis. New York: Pitman, Zbl. 529.30001.MATHGoogle Scholar
  5. Burkholder, D. L. (1979a): Martingale theory and harmonic analysis on Euclidean spaces. In: Harmonic analysis on Euclidean spaces, Proc. Symp. Pure Math. 35, part II, 283–301. Providence, RI: Am. Math. Soc, Zbl. 417.60055.MathSciNetCrossRefGoogle Scholar
  6. Burkholder, D. L. (1979b): A sharp inequality for martingale transforms. Ann. Probab. 7, No. 4, 858–863, Zbl. 416.60047.MathSciNetMATHCrossRefGoogle Scholar
  7. Burkholder, D. L. and Gundy, R. F. (1972): Distribution function inequalities for the area integral. Stud. Math. 44, No. 6, 527–544, Zbl. 219,179.MathSciNetMATHGoogle Scholar
  8. Calderón, A. P. (1965): Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099, Zbl. 151,169.MATHCrossRefGoogle Scholar
  9. Calderón, A. P. (1976): Inequalities for the maximal function relative to a metric. Stud. Math. 57, 297–306, Zbl. 341.44007.MATHGoogle Scholar
  10. Calderón, A. P. (1977): Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74, 1324–1327, Zbl. 373.44003.MATHCrossRefGoogle Scholar
  11. Calderón, A. P. (1980): Commutators, singular integrals on Lipschitz curves and applications. In: Proc. Int. Congr. Math., Helsinki 1978, vol. 1, 85–96, Zbl. 429.35077.Google Scholar
  12. Calderón, A. P. and Torchinsky, A. (1975/77): Parabolic maximal functions associated with a distribution. Adv. Math. 16, 1–64, Zbl. 315.46037; 24, 101–171, Zbl. 355.46021.Google Scholar
  13. Carleson, L. (1967): Selected problems on exceptional sets. Princeton: Van Nostrand. (98 pp.), Zbl. 189,109.MATHGoogle Scholar
  14. Chang, S. Y. A. and Fefferman, R. (1980): A continuous version of the duality of H 1 with BMO on the bidisk. Ann. Math., II. Ser. 112, 179–201, Zbl. 451.42014.MathSciNetMATHCrossRefGoogle Scholar
  15. Chang, S. Y. A. and Fefferman, R. (1985): Some recent developments in Fourier analysis and H p theory on product domains. Bull. Am. Math. Soc, New Ser. 12, 1–43, Zbl. 557.42007.MathSciNetMATHCrossRefGoogle Scholar
  16. Coifman, R., David, G., and Meyer, Y. (1983): La solution des conjectures de Calderön. Adv. Math. 48, No. 2, 144–148, Zbl. 518.42024.MathSciNetMATHCrossRefGoogle Scholar
  17. Coifman, R., Mcintosh, A., and Meyer, Y. (1982): L’intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes Lipschitziennes. Ann. Math., II. Ser. 116, No. 2, 361–387, Zbl. 497.42012.MathSciNetMATHCrossRefGoogle Scholar
  18. Coifman, R. and Meyer, Y. (1975): On commutators on singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331, Zbl. 324.44005.MathSciNetMATHCrossRefGoogle Scholar
  19. Coifman, R. and Meyer, Y. (1978): Au-delà des opérateurs pseudo-differentiels. Asterisque 57 (188 pp.), Zbl. 483.35082.MathSciNetGoogle Scholar
  20. Coifman, R., Meyer, Y., and Stein, E. M. (1983): Un nouvel espace fonctionnel adaptéà l’étude des opérateurs définis par des intégrales singulières. In: Harmonie Analysis, Proc. Conf. Cortona/Italy 1982, Lect. Notes Math. 992, 1–15, Zbl. 523.42016.MathSciNetCrossRefGoogle Scholar
  21. Coifman, R., Meyer, Y., and Stein, E. M. (1985): Some new function spaces and their applications to harmonic analysis. J. Funct. Anal. 62, 304–335, Zbl. 569.42016.MathSciNetMATHCrossRefGoogle Scholar
  22. Coifman, R. and Weiss, G. (1971): Analyse harmonique non-commutative sur certains espaces homogénes. Lect. Notes Math. 242. (160 pp.), Zbl. 224.43006.MathSciNetGoogle Scholar
  23. Coifman, R. and Weiss, G. (1977): Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645, Zbl. 358.30023.MathSciNetMATHCrossRefGoogle Scholar
  24. Coifman, R. and Weiss, G. (1978): Book review of “Littlewood-Paley and multiplier theory” by R. E. Edwards and G. I. Gaudry. Bull. Am. Math. Soc. 84, 242–250.MathSciNetCrossRefGoogle Scholar
  25. Cowling, M. G. (1981): On Littlewood-Paley-Stein theory. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 1, 21–55, Zbl. 472.42012.MathSciNetMATHGoogle Scholar
  26. Dahlberg, B. (1977): Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, 275–288, Zbl. 406.28009.MathSciNetMATHCrossRefGoogle Scholar
  27. Dahlberg, B. (1980): Weighted norm inequalities for the Luzin area integrals and the nontangential maximal functions for functions harmonic in a Lipschitz domain. Stud. Math. 67, 297–314, Zbl. 449.31002.MathSciNetMATHGoogle Scholar
  28. Dahlberg, B. and Kenig, C. (1985): Hardy spaces and the Neumann problem in LP for Laplace equation in Lipschitz domains I. Ann. Math., II. Ser. 125, No. 3, 437–465, Zbl. 658.35027.MathSciNetCrossRefGoogle Scholar
  29. Daubechies, I., Grossmann, A., and Meyer, Y. (1986): Painless non-orthogonal expansions. J. Math. Phys. 27, 1271–1283, Zbl. 608.46014.MathSciNetMATHCrossRefGoogle Scholar
  30. David, G. (1984): Opérateurs intégraux singuliers sur certains courbes du plan complexe. Ann. Sci. Ec. Norm. Super., IV. Ser. 17, No. 1, 157–189, Zbl. 537.42016.MATHGoogle Scholar
  31. David, G. (1986): Noyau de Cauchy et opérateurs de Calderón-Zygmund. Thèses d’Etat, Univ. Paris-Sud (210 pp.).Google Scholar
  32. David, G. (1987): Opérateurs d’intégrales singulières sur les surfaces regulières. Preprint, École Polytechnique (49 pp.). Appeared in: Ann. Sci. Ec. Norm. Super., IV Ser. 21, No. 2, 225–258, (1988), Zbl. 655.42013.Google Scholar
  33. David, G. and Journé, J.-L. (1984): A boundedness criterion for generalized Calderon-Zygmund operators. Ann. Math., II. Ser. 120, No. 2, 371–397, Zbl. 567.47025.MATHCrossRefGoogle Scholar
  34. David, G., Journé, J.-L., and Semmes, S. (1985): Opérateurs de Calderon-Zygmund, fonctions para-accretive et interpolation. Rev. Mat. Iberoam. 1, No. 4, 1–56, Zbl. 604.42014.MATHCrossRefGoogle Scholar
  35. Durrett, R. (1984): Brownian motion and martingales in analysis. Belmont, California: Wadsworth (328 pp.), Zbl. 554.60075.MATHGoogle Scholar
  36. Dyn’kin, E. M. (1987): Methods of the theory of singular integrals (Hilbert transform and Calderón-Zygmund theory). In: Itogi Nauki Tekh, Ser. Sovrem. Probl. Mat. 15, 197–202. Moscow: VINITI. English translation: Encyclopaedia Math. Sci. Vol. 15, pp. 167–259. Berlin Heidelberg: Springer-Verlag (1991), Zbl. 661.42009.Google Scholar
  37. Dyn’kin, E. M. (1981): A constructive characterization of Sobolev and Besov classes. Tr. Mat. Inst. Steklova 155, 41–76. English translation: Proc. Steklov Inst. Math. 155, 39–74 (1983), Zbl. 496.46021.MathSciNetGoogle Scholar
  38. Fabes, E., Jerison, D., and Kenig, C. (1982): Multilinear Littlewood-Paley estimates with applications to partial differential equations. Proc. Natl. Acad. Sci. USA 79, 5746–5750, Zbl. 501.35014.MathSciNetMATHCrossRefGoogle Scholar
  39. Fabes, E., Jodeit, M. jun., and Lewis, T. (1977): Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 26, 95–114, Zbl. 363.35010.MathSciNetMATHCrossRefGoogle Scholar
  40. Fabes, E., Jodeit, M. jun., and Rivière, N. (1978): Potential theoretic techniques for boundary value problems on C 1-domains. Acta Math. 141, 165–186, Zbl. 402.31009.MathSciNetMATHCrossRefGoogle Scholar
  41. Fefferman, Ch. (1972): The multiplier problem for the ball. Ann. Math., II. Ser. 94, 330–336, Zbl. 234.42009.MathSciNetCrossRefGoogle Scholar
  42. Fefferman, Ch. and Stein, E. M. (1972): H p spaces of several variables. Acta Math. 129, 137–193, Zbl. 257.46078.MathSciNetMATHCrossRefGoogle Scholar
  43. Folland, G. B. and Stein, E. M. (1982), Hardy spaces on homogeneous groups. (Mathematical Notes 28.) Princeton: Univ. Press (284 pp.), Zbl. 508.42025.MATHGoogle Scholar
  44. Garnett, J. B. (1981): Bounded analytic functions. New York: Academic Press (467 pp.), Zbl. 469.30024.MATHGoogle Scholar
  45. Garsia, A. (1973): Martingale inequalities. Reading, Mass.: Benjamin (184 pp.), Zbl. 284.60046.MATHGoogle Scholar
  46. Garcia-Cuerva, J. L. and Rubio de Francia, J. (1985): Weigthed norm inequalities and related topics. Amsterdam New York Oxford: North-Holland (604 pp.), Zbl. 578.46046.Google Scholar
  47. Gelbaum, B. R. and Olmsted, I. M. (1964): Counterexamples in analysis. San Francisco, CA: Holden-Day (194 pp.), Zbl. 121,289.Google Scholar
  48. Gikhman, I. I. and Skorokhod, A. V. (1982): Stochastic differential equations and their applications. Kiev: Naukova Dumka. [Russian], Zbl. 557.60041.MATHGoogle Scholar
  49. Goluzin, G. M. (1966): Geometric theory of functions of a complex variable. Moscow: Nauka (628 pp.). English translation: Translation of Mathematical Monographs 26. Providence, RI: Amer. Math. Soc. (1969), Zbl. 148,306.MATHGoogle Scholar
  50. Gundy, R. F. (1980): Inégalités pour martingales à un et deux indices: l’espace H p . In: Ecole d’été de probabilites de Saint-Fleur VIII — 1978, Lect. Notes Math. 774, 251–334, Zbl. 427.60046.MathSciNetGoogle Scholar
  51. Hunt, R. and Wheeden, R. (1968): On the boundary values of harmonie functions. Trans. Am. Math. Soc. 132, No. 2, 307–322, Zbl. 159,405.MathSciNetMATHCrossRefGoogle Scholar
  52. Jerison, D. and Kenig, C. (1980): An identity with applications to harmonic measure. Bull. Am. Math. Soc, New Ser. 2, No. 3, 447–451, Zbl. 436.31002.MathSciNetMATHCrossRefGoogle Scholar
  53. Jerison, D. and Kenig, C. (1982a): Boundary behavior of harmonie functions in nontangentially accesible domains. Adv. Math. 46, No. 3, 80–147, Zbl. 514.31003.MathSciNetMATHCrossRefGoogle Scholar
  54. Jerison, D. and Kenig, C. (1982b): Hardy spaces, (A)8 and singular integrals on chord arc domains. Math. Scand. 50, 221–247, Zbl. 509.30025.MathSciNetMATHGoogle Scholar
  55. Journé, J.-L. (1983): Calderón-Zygmund operators, pseudo-differential operators and the Cauchy integral of Calderón. Lect. Notes Math. 994, (128 pp.), Zbl. 508.42021.Google Scholar
  56. Koosis, P. (1980): Introduction to H p spaces. Cambridge: Univ. Press (376 pp.), Zbl. 435.30001.MATHGoogle Scholar
  57. Kurtz, D. (1980): Littlewood-Paley and multiplier theory on weighted L p spaces. Trans. Am. Math. Soc. 259, No. 1, 235–254, Zbl. 436.42012.MathSciNetMATHGoogle Scholar
  58. Larsen, R. (1971): An introduction to the theory of multipliers. (Grundlehren Math. Wiss. 175). Berlin Heidelberg: Springer-Verlag (282 pp.), Zbl. 213,133.MATHCrossRefGoogle Scholar
  59. Lemarié, P. G. and Meyer, Y. (1986): Ondelettes et bases hilbertiennes. Rev. Mat. Iberoam. 2, 1–18, Zbl. 657.42028.CrossRefGoogle Scholar
  60. Littlewood, J. E. and Paley, R. E. A. C. (1931/36): Theorems on Fourier series and power series. J. Lond. Math. Soc. 6, 230–233, Zbl. 2,188; Proc. Lond. Math. Soc, II. ser. 42, 52–89, Zbl. 15,254; 43, 105–126, Zbl. 16,301.CrossRefGoogle Scholar
  61. Macias, R. and Segovia, C. (1979a): Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270, Zbl. 431.46018.MathSciNetMATHCrossRefGoogle Scholar
  62. Macias, R. and Segovia, C. (1979b): A decomposition into atoms of distributions on spaces of homogeneous type. Adv. Math. 33, 271–309, Zbl. 431.46019.MathSciNetMATHCrossRefGoogle Scholar
  63. Macias, R. and Segovia, C. (1979c): Singular integrals on generalized Lipschitz and Hardy spaces. Stud. Math. 65, No. 1, 55–75, Zbl. 479.42014.MathSciNetMATHGoogle Scholar
  64. McIntosh, A. and Meyer, Y. (1985): Algèbres d’opérateurs définis par les intégrales singulières. C. R. Acad. Sci., Paris, Ser. I 301, No. 8, 395–397, Zbl. 584.47030.MathSciNetMATHGoogle Scholar
  65. Maz’ya, V. G. (1985): Sobolev spaces. Leningrad: LGU (416 pp.). English translation: Berlin Heidelberg: Springer-Verlag (1985), Zbl. 692.46023.Google Scholar
  66. Meyer, Y. (1985): Continuité sur les espaces de Holder et de Sobolev des opérateurs définis par les intégrales singulières. In: Recent progress in Fourier analysis, Proc. Semin., El Escorial/Spain 1983, North-Holland Math. Stud. 111, 145–172, Zbl. 616.42008.Google Scholar
  67. Meyer, Y. (1987): Wavelets and operators. Preprint CEREMADE No 8704, Univ. Paris-Dauphine (108 pp.). Appeared in: Lond. Math. Soc. Lect. Note Ser. 137, 256–365 (1989).Google Scholar
  68. Muckenhoupt, B. and Wheeden, R. (1974): Norm inequalities for the Littlewood-Paley function (math). Trans. Am. Math. Soc. 191, 95–111, Zbl. 289.44005.MathSciNetMATHGoogle Scholar
  69. Murai, T. (1983): Boundedness of singular integrals of Calderón type. Proc. Japan Acad., Ser. A 59, No. 8, 364–367, Zbl. 542.42008.MathSciNetMATHCrossRefGoogle Scholar
  70. Murai, T. and Tschamitchian, P. (1984): Boundedness of singular integral operators of Calderon type V, VI. Preprint series, College of general education, Nagoya, No. 8; No. 12. Appeared in: Adv. Math. 59, 71–81, Zbl. 608.42010.CrossRefGoogle Scholar
  71. Peetre, J. (1976): New thoughts on Besov spaces. (Duke Univ. Math. Series 1.) Durham, N. C: Math. Department, Duke Univ. (304 pp.), Zbl. 356.46038.MATHGoogle Scholar
  72. Pommerenke, Ch. (1978): Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment. Math. Helv. 52, No. 4, 591–602, Zbl. 369.30012.MathSciNetGoogle Scholar
  73. Privalov, I. I. (1950): Boundary values of analytic functions. Moscow Leningrad: GITTL (336 pp.), Zbl. 41,397.Google Scholar
  74. Riesz, F. and Sz.-Nagy, B. (1972): Leçons d’analyse fonctionelle. Budapest: Akad. Kiadó English translation: New York: Frederick Ungar 1978 (467 pp.), Zbl. 46,331; Zbl. 64,354.Google Scholar
  75. Rudin, W. (1973): Functional analysis. New York: McGraw-Hill (432 pp.), Zbl. 253.46001.MATHGoogle Scholar
  76. Rvachev, V. A. (1986): Atomary functions and their applications. In: The theory of R-functions and contemporary problems of applied mathematics, pp. 45–65. Kiev: Naukova Dumka [Russian].Google Scholar
  77. Stein, E. M. (1970a): Singular integrals and differentiability properties of functions. Princeton: Univ. Press (290 pp.), Zbl. 207,135.MATHGoogle Scholar
  78. Stein, E. M. (1970b): Topics in harmonic analysis related to the Littlewood-Paley theory. Princeton: Univ. Press (146 pp.), Zbl. 193,105.MATHGoogle Scholar
  79. Stein, E. M. (1982): The development of square functions in the work of A. Zygmund. Bull. Am. Math. Soc, New Ser. 7, 359–376, Zbl. 526.01021.MATHCrossRefGoogle Scholar
  80. Strichartz, R. S. (1967): Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060, Zbl. 145,383.MathSciNetMATHGoogle Scholar
  81. Strömberg, J.-O. and Torchinsky, A. (1980): Weights, sharp maximal functions and Hardy spaces. Bull. Am. Math. Soc, New Ser. 3, 1053–1056, Zbl. 452.43004.MATHCrossRefGoogle Scholar
  82. Triebel, H. (1978): Interpolation theory, function spaces, differential operators. Berlin: VEB Wiss. Verlag (528 pp.) and North-Holland Publ. Co., Zbl. 387.46032.Google Scholar
  83. Triebel, H. (1983): Theory of function spaces. Basel: Birkhäuser (432 pp.), Zbl. 546.46027.CrossRefGoogle Scholar
  84. Uchiyama, A. (1980): A maximal function characterization of H p on the space of homogeneous type. Trans. Am. Math. Soc. 262, 579–592, Zbl. 503.46020.MathSciNetMATHGoogle Scholar
  85. Varopoulos, N. T. (1980): Aspects of probabilistic Littlewood-Paley theory. J. Funct. Anal. 38, No. 1, 25–60, Zbl. 462.60050.MathSciNetMATHCrossRefGoogle Scholar
  86. Verchota, G. (1984): Layer potentials and regularity for the Dirichlet problem for Laplace’s operator in Lipschitz domains. J. Funct. Anal. 59, No. 3, 572–611, Zbl. 589.31005.MathSciNetMATHCrossRefGoogle Scholar
  87. Vol’berg, A. P. and Konyagin, S. V. (1984): On every compact set in ℝn there exists a homogeneous measure. Dokl. Akad. Nauk SSSR 278, No. 3, 783–786. English translation: Sov. Math., Dokl. 30, 453–456 (1984), Zbl. 598.28010.MathSciNetGoogle Scholar
  88. Wittmann, R. (1987): Application of a theorem of M. G. Kreĭn to singular integrals. Trans. Am. Math. Soc. 299, No. 2, 581–599, Zbl. 596.42005.MathSciNetMATHGoogle Scholar
  89. Zygmund, A. (1959): Trigonometric series I-II. Cambridge: Univ. Press (383 pp.; 354 pp.), Zbl. 85,56.Google Scholar
  90. Zygmund, A. (1979): Harmonic analysis on Euclidean spaces, Part I-II, Proc Symp. Pure Math. 35. Providence, RI: Amer. Math. Soc. (898 pp.), Zbl. 407.00005, Zbl. 407.00006.Google Scholar
  91. Zygmund, A. (1982): Harmonic analysis, Proc. Minneapolis, 1981, Lect. Notes Math. 908. Berlin: Springer-Verlag (326 pp.), Zbl. 471.00014.Google Scholar
  92. Zygmund, A. (1983): Harmonic analysis. Proc. Conf. Cortona/Italy, 1982, Lect. Notes Math. 992. Berlin: Springer-Verlag (450 pp.), Zbl. 504.00013.Google Scholar
  93. Zygmund, A. (1985): Recent progress in Fourier analysis (eds. I. Peral and J. L. Rubio de Francia). Amsterdam: North-Holland (268 pp.), Zbl. 581.00009.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • E. M. Dyn’kin

There are no affiliations available

Personalised recommendations