Skip to main content

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

  • Chapter
Commutative Harmonic Analysis IV

Part of the book series: Encyclopaedia of Mathematical Sciences ((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)).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Battle, G. and Federbush, P. (1982): A phase cell cluster expansion for Euclidean field theories. Ann. Phys. 142, 95–139.

    Article  MathSciNet  Google Scholar 

  • Belinskiĭ, P. P. (1974): General properties of quasiconformal mappings. Moscow: Nauka (98 pp.) [Russian] Zbl. 292.30019.

    Google Scholar 

  • Brackx, F., Delange, R., and Sommer, F. (1982): Clifford analysis. New York: Pitman, Zbl. 529.30001.

    MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Burkholder, D. L. (1979b): A sharp inequality for martingale transforms. Ann. Probab. 7, No. 4, 858–863, Zbl. 416.60047.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Calderón, A. P. (1965): Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 53, 1092–1099, Zbl. 151,169.

    Article  MATH  Google Scholar 

  • Calderón, A. P. (1976): Inequalities for the maximal function relative to a metric. Stud. Math. 57, 297–306, Zbl. 341.44007.

    MATH  Google Scholar 

  • Calderón, A. P. (1977): Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 74, 1324–1327, Zbl. 373.44003.

    Article  MATH  Google Scholar 

  • 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 

  • 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 

  • Carleson, L. (1967): Selected problems on exceptional sets. Princeton: Van Nostrand. (98 pp.), Zbl. 189,109.

    MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Coifman, R., David, G., and Meyer, Y. (1983): La solution des conjectures de Calderön. Adv. Math. 48, No. 2, 144–148, Zbl. 518.42024.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Coifman, R. and Meyer, Y. (1978): Au-delà des opérateurs pseudo-differentiels. Asterisque 57 (188 pp.), Zbl. 483.35082.

    MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Coifman, R. and Weiss, G. (1971): Analyse harmonique non-commutative sur certains espaces homogénes. Lect. Notes Math. 242. (160 pp.), Zbl. 224.43006.

    MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Cowling, M. G. (1981): On Littlewood-Paley-Stein theory. Rend. Circ. Mat. Palermo, II. Ser., Suppl. 1, 21–55, Zbl. 472.42012.

    MathSciNet  MATH  Google Scholar 

  • Dahlberg, B. (1977): Estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, 275–288, Zbl. 406.28009.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  Google Scholar 

  • Daubechies, I., Grossmann, A., and Meyer, Y. (1986): Painless non-orthogonal expansions. J. Math. Phys. 27, 1271–1283, Zbl. 608.46014.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • David, G. (1986): Noyau de Cauchy et opérateurs de Calderón-Zygmund. Thèses d’Etat, Univ. Paris-Sud (210 pp.).

    Google Scholar 

  • 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 

  • 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.

    Article  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Durrett, R. (1984): Brownian motion and martingales in analysis. Belmont, California: Wadsworth (328 pp.), Zbl. 554.60075.

    MATH  Google Scholar 

  • 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 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Fefferman, Ch. (1972): The multiplier problem for the ball. Ann. Math., II. Ser. 94, 330–336, Zbl. 234.42009.

    Article  MathSciNet  Google Scholar 

  • Fefferman, Ch. and Stein, E. M. (1972): H p spaces of several variables. Acta Math. 129, 137–193, Zbl. 257.46078.

    Article  MathSciNet  MATH  Google Scholar 

  • Folland, G. B. and Stein, E. M. (1982), Hardy spaces on homogeneous groups. (Mathematical Notes 28.) Princeton: Univ. Press (284 pp.), Zbl. 508.42025.

    MATH  Google Scholar 

  • Garnett, J. B. (1981): Bounded analytic functions. New York: Academic Press (467 pp.), Zbl. 469.30024.

    MATH  Google Scholar 

  • Garsia, A. (1973): Martingale inequalities. Reading, Mass.: Benjamin (184 pp.), Zbl. 284.60046.

    MATH  Google Scholar 

  • 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 

  • Gelbaum, B. R. and Olmsted, I. M. (1964): Counterexamples in analysis. San Francisco, CA: Holden-Day (194 pp.), Zbl. 121,289.

    Google Scholar 

  • Gikhman, I. I. and Skorokhod, A. V. (1982): Stochastic differential equations and their applications. Kiev: Naukova Dumka. [Russian], Zbl. 557.60041.

    MATH  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Koosis, P. (1980): Introduction to H p spaces. Cambridge: Univ. Press (376 pp.), Zbl. 435.30001.

    MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Larsen, R. (1971): An introduction to the theory of multipliers. (Grundlehren Math. Wiss. 175). Berlin Heidelberg: Springer-Verlag (282 pp.), Zbl. 213,133.

    Book  MATH  Google Scholar 

  • Lemarié, P. G. and Meyer, Y. (1986): Ondelettes et bases hilbertiennes. Rev. Mat. Iberoam. 2, 1–18, Zbl. 657.42028.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • Macias, R. and Segovia, C. (1979a): Lipschitz functions on spaces of homogeneous type. Adv. Math. 33, 257–270, Zbl. 431.46018.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • Macias, R. and Segovia, C. (1979c): Singular integrals on generalized Lipschitz and Hardy spaces. Stud. Math. 65, No. 1, 55–75, Zbl. 479.42014.

    MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Maz’ya, V. G. (1985): Sobolev spaces. Leningrad: LGU (416 pp.). English translation: Berlin Heidelberg: Springer-Verlag (1985), Zbl. 692.46023.

    Google Scholar 

  • 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 

  • 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 

  • Muckenhoupt, B. and Wheeden, R. (1974): Norm inequalities for the Littlewood-Paley function (math). Trans. Am. Math. Soc. 191, 95–111, Zbl. 289.44005.

    MathSciNet  MATH  Google Scholar 

  • Murai, T. (1983): Boundedness of singular integrals of Calderón type. Proc. Japan Acad., Ser. A 59, No. 8, 364–367, Zbl. 542.42008.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    MATH  Google Scholar 

  • Pommerenke, Ch. (1978): Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation. Comment. Math. Helv. 52, No. 4, 591–602, Zbl. 369.30012.

    MathSciNet  Google Scholar 

  • Privalov, I. I. (1950): Boundary values of analytic functions. Moscow Leningrad: GITTL (336 pp.), Zbl. 41,397.

    Google Scholar 

  • 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 

  • Rudin, W. (1973): Functional analysis. New York: McGraw-Hill (432 pp.), Zbl. 253.46001.

    MATH  Google Scholar 

  • 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 

  • Stein, E. M. (1970a): Singular integrals and differentiability properties of functions. Princeton: Univ. Press (290 pp.), Zbl. 207,135.

    MATH  Google Scholar 

  • Stein, E. M. (1970b): Topics in harmonic analysis related to the Littlewood-Paley theory. Princeton: Univ. Press (146 pp.), Zbl. 193,105.

    MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • Strichartz, R. S. (1967): Multipliers on fractional Sobolev spaces. J. Math. Mech. 16, 1031–1060, Zbl. 145,383.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MATH  Google Scholar 

  • 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 

  • Triebel, H. (1983): Theory of function spaces. Basel: Birkhäuser (432 pp.), Zbl. 546.46027.

    Book  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Varopoulos, N. T. (1980): Aspects of probabilistic Littlewood-Paley theory. J. Funct. Anal. 38, No. 1, 25–60, Zbl. 462.60050.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Zygmund, A. (1959): Trigonometric series I-II. Cambridge: Univ. Press (383 pp.; 354 pp.), Zbl. 85,56.

    Google Scholar 

  • 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 

  • Zygmund, A. (1982): Harmonic analysis, Proc. Minneapolis, 1981, Lect. Notes Math. 908. Berlin: Springer-Verlag (326 pp.), Zbl. 471.00014.

    Google Scholar 

  • Zygmund, A. (1983): Harmonic analysis. Proc. Conf. Cortona/Italy, 1982, Lect. Notes Math. 992. Berlin: Springer-Verlag (450 pp.), Zbl. 504.00013.

    Google Scholar 

  • 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 

Download references

Authors

Editor information

V. P. Khavin N. K. Nikol’skiǐ

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Dyn’kin, E.M. (1992). Methods of the Theory of Singular Integrals: Littlewood-Paley Theory and Its Applications. In: Khavin, V.P., Nikol’skiǐ, N.K. (eds) Commutative Harmonic Analysis IV. Encyclopaedia of Mathematical Sciences, vol 42. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-06301-9_2

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-06301-9_2

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-08103-3

  • Online ISBN: 978-3-662-06301-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics