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Linear and Complex Analysis Problem Book pp 308–352Cite as

Singular integrals, BMO, Hp

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1043)

Keywords

  • Riemann Surface
  • Hardy Space
  • Toeplitz Operator
  • Blaschke Product
  • Bloch Function

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References

  1. Calderón A.P. On the Cauchy integral on Lipschitz curves and related operators.-Proc.N.Ac.Sc. 1977, 4, 1324–27.

    CrossRef  MATH  Google Scholar 

  2. Coifman R.R., Meyer Y. Commutateurs d'integrales singulières et opérateurs multilinéaires.-Ann. Inst. Fourier (Grenoble), 1978, 28, N 3, xi, 177–202.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Coifman R.R., Meyer Y. Multilinear pseudo-differential operators and commutators, to appear.

    Google Scholar 

References

  1. Хведелидэе Б.В. Метод интегралов типа Кощ в раэрывных граничных эадачах теории голоморфных функций одной комплексной переменной. "Современные проблемы математики", т.7, Москва, 1975, 5–162.

    Google Scholar 

  2. Данилюк И.И. Нерегулярные граничные эадачи на плоскости, Москва, Наука, 1975.

    Google Scholar 

  3. Duren P.L., Shapiro H.S., Shields A.L. Singular measures and domains not of Smirnov type.-Duke Math. J., 1966, v.33, N 2, 247–254.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Тумаркин Г.Ц. Граничные свойства аналитических функций, представимых интегралами типа Кощи.-Матем.сб., 1971, 84 (126), No 3, 425–439.

    Google Scholar 

  5. Паатащвили В.А. О сингулярных интегралах Кощи.-Сообш. АН Груэ.ССР, 1969, 53, No 3, 529–532.

    Google Scholar 

  6. Дынькин Е.М. О равномерном приближении многочленами в комплексной плоскости.-Зап.научн.семин.ЛОМИ, 1975, 56, 164–165.

    Google Scholar 

  7. Дынькин Е.М. О равномерном приближении функции в жорда-новых областях.-Сиб.мат.ж. 1977, 18, No 4, 775–786.

    Google Scholar 

  8. Andersson Jan-Erick, Ganelius Tord. The degree of approximation by rational function with fixed poles.-Math.Z., 1977, 153, N 2, 161–166.

    MathSciNet  CrossRef  MATH  Google Scholar 

  9. Тумаркин Г.Ц. Граничные свойства конформных отображений некоторых классов областей.-сб."Некоторые вопросы современной теории функций", Новосибирск, 1976, 149–160.

    Google Scholar 

  10. Альфорс Л. Лекции по кваэиконформным отображениям. Москва, Мир, 1969.

    Google Scholar 

  11. Хавин В.П. Граничные свойства интегралов типа Кощи и гармонически сопряженных функций в областях со спрямляемой границей.-Матем.сб., 1965, 68(110), 499–517.

    Google Scholar 

  12. Белый В.И., Миклюко в В.М. Некоторые свойства конформных и кваэиконформных отображений и прямые теоремы конструктивной теории функций.-Иэв.АН СССР, серия матем.,1974, No 6, 1343–1361.

    Google Scholar 

  13. Белый В.И. Конформные отображения и приближение аналитических функций в областях с кваэиконформной границей.-Мат.сб., 1977, 102, No 3, 331–361.

    Google Scholar 

References

  1. Cotlar M., Sadosky C. On some LP versions of the Helson-Szegö theorem.-In: Conference on Harmonic Analysis in Honor of Antoni Zygmund, Wadsworth, Belmont, California, 1983, 306–317.

    Google Scholar 

  2. Jurkat W.B., Sampson G. On rearrangement and weight inequalities for the Fourier transform, to appear.

    Google Scholar 

  3. Muckenhoupt B. Weighted norm inequalities for classical operators.-Proc.Symp. in Pure Math 35 (1), 1979, 69–83.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Muckenhoupt B. Weighted norm inequalities for the Fourier transform.-Trans.Amer.Math.Soc., to appear.

    Google Scholar 

  5. Sawyer E. Two weight norm inequalities for certain maximal and integral operators. In: Harmonic Analysis, Lecture Notes Math. 908, Springer, Berlin 1982, 102–127.

    CrossRef  Google Scholar 

  6. Sawyer E. Norm inequalities relating singular integrals and the maximal function, to appear.

    Google Scholar 

  7. Sawyer E. Weighted norm inequalities for the n-dimensional Hardy operator, to appear.

    Google Scholar 

Reference

  1. Coifman R.R., McIntosh A., Meyer Y. L'intégrale de Cauchy définit un opérateur borné sur L2 pour les courbes Lipschitziennes.-Ann.Math., 1982, 116, 361–387.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Киоляков С.В. Козффициенты Фурье граничных эначений функций, аналитических в круге и в бидиоке.-Труда Матем.ин-та вы. В.А.Стеклова, 1981, 155, 77–94.

    Google Scholar 

  2. Bourgain J. Extensions of H∞-valued functions and bounded bianalytic functions. Preprint, 1982.

    Google Scholar 

References

  1. Гохберг И.Ц., Крупник Н.Я. О норме преобраэования Гильберта в пространстве LP.-Функц.анал. и его прил., 1968, 2, No 2, 91–92.

    Google Scholar 

  2. Pichorides S.K. On the best values of the constants in in the theorems of M.Riesz, Zygmund and Kolmogorov.-Studia Math., 1972, 44, N 2, 165–179.

    MathSciNet  MATH  Google Scholar 

  3. Крупник Н.Я., Полонский Е.П. О норме оператора сингулярного интегрирования.-Функц.анал. и его прил., 1975, 9, No 4, 73–74.

    Google Scholar 

  4. Вербицкий И.Э. Оценка нормы функции иэ пространства Харди череэ норму ее вешественной и мнимой части.-В сб."Матем. исследования", Кищинев, Щтиинца, 1980, No 54, 16–20.

    Google Scholar 

  5. Вербицкий И.Э., Крупник Н.Я. Точные константы в теоремах К.И.Бабенко и Б.В.Хведелидэе об ограниченности сингулярного оператора.-Сообш.АН Груэ.ССР, 1977, 85, No 1, 21–24.

    Google Scholar 

  6. Гохберг И.Ц., Крупник Н.Я. Введение в теорию сингулярных интегральных операторов.-Кищинев, Щтвднца, 1973.

    Google Scholar 

  7. Никольский Н.К. Лекции об операторе сдвига. М.: Наука, 1980.

    Google Scholar 

  8. Hardy G.H., Littlewood J.E., Pólya G. Inequalities. 2nd ed. Cambridge Univ. Press, London and New York, 1952.

    MATH  Google Scholar 

References

  1. David G. Courbes corde-arc et espaces de Hardy généralisés.-Ann.Inst.Fourier (Grenoble), 1982, 32, 227–239.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Garnett J., O'Farrell A. Sobolev approximation by a sum of subalgebras on the circle.-Pacific J.Math. 1976, 65, 55–63.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Jones P. Homeomorphisms of the line which preserve BMO, to appear in Arkiv för Matematik.

    Google Scholar 

References

  1. Arocena R. A refinement of the Helson-Szegö theorem and the determination of the extremal measures.-Studia Math, 1981, LXXI, 203–221.

    MathSciNet  MATH  Google Scholar 

  2. Coifman R., Rochberg R. Projections in weighted spaces, skew projections, and inversion of Toeplitz operators.-Integral Equations and Operator Theory, 1982, 5, 145–159.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Coifman R., Rochberg R., Weiss G. Factorization Theorems for Hardy Spaces in Several Variables.-Ann. Math. 1976, 103, 611–635.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Marshall D. Blaschke products generate H∞.-Bull.Amer. Math.Soc., 1976, 82, 494–496.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Marshall D. Subalgebras of L∞ containing H∞.-Acta Math., 1976, 137, 91–98.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Hunt R.A., Muckenhoupt B., Wheeden R.L. Weighted norm inequalities for the conjugate function and Hilbert transform.-Trans.Amer.Math.Soc., 1973, 176, 227–251.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Helson H., Szegö G. A problem in prediction theory.-Ann.Math.Pure Appl., 1960, 51, 107–138.

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Fefferman C., Stein E.M. Hp spaces of several variables.-Acta Math., 1972, 129, 137–193.

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Carleson L. Two remarks on H1 and BMO.-Advances in Math., 1976, 22, 269–277.

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Stein E.M. Singular integrals and differentiability properties of functions. Princeton N.J., 1970.

    Google Scholar 

  8. Janson S. Characterization of H1 by singular integral transforms on martingales and Rn.-Math.Scand., 1977, 41, 140–152.

    MathSciNet  MATH  Google Scholar 

References

  1. Wolff T. Counterexamples to two variants of the Helson-Szegö theorem. Preprint, 1983, Institut Mittag-Leffler, 11.

    Google Scholar 

  2. Jones P. Carleson measures and the Fefferman — Stein decomposition of BMO (ℝ).-Ann. of Math., 1980, 111, 197–208.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Jones P. L∞-estimates for the -problem. To appear in Acta Math.

    Google Scholar 

  4. Uchiyama A. A constructive proof of the Fefferman — Stein decomposition of BMO (ℝn).-Acta Math., 1982, 148, 215–241.

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Stray A. Two applications of the Schur — Nevanlinna algorithm.-Pacif. J. of Math., 1980, 91, N 1, 223–232.

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Rubio de Francia J.L. Factorization and extrapolation by weights.-Bull.Amer.Math.Soc., 1982, 7, N 2, 393–395.

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Amar E. Représentation des fonctions de BMO et solutions de l'équation 6. Preprint, 1978, Univ. Paris XI Orsay.

    Google Scholar 

  8. Coifman R., Jones P.W., Rubio de Francia J.L. Constructive decomposition of BMO functions and factorization of Ap weights.-Proc.Amer.Math.Soc., 1983, 87, N 4, 675–680.

    MathSciNet  MATH  Google Scholar 

References

  1. Baernstein A. II. Univalence and bounded mean oscillation.-Mich.Math.J., 1976, 23, 217–223.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Helson H., Szegö G. A problem in prediction theory.-Ann.Mat.Pura Appl., 1960, 51, (4), 107–138.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Hunt R., Muckenhoupt B., Wheeden R. Weighted norm inequalities for the conjugate function and Hilbert transform.-Trans.Amer.Math.Soc., 1973, 176, 227–251.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Fefferman C., Stein E.M. Hp spaces of several variables.-Acta Math., 1972, 129, 137–193.

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Baernstein A. II. Integral means, univalent functions and circular symmetrization.-Acta Math., 1974, 133, 139–169.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Zygmund A. Trigonometric series, vol.I. Cambridge, Cambridge Univ.Press. 1959.

    MATH  Google Scholar 

  2. Anderson J.M., Clunie J., Pommerenke Ch. On Bloch functions and normal functions.-J.Reine Angew.Math. 1974, 270, 12–37.

    MathSciNet  MATH  Google Scholar 

  3. Pommerenke Ch. On univalent functions, Bloch functions and VMOA.-Math.Ann., 1978, 236, N 3, 199–208.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Sarason D. Functions of vanishing mean oscillation.-Trans.Amer.Math.Soc. 1975, 207, 391–405.

    MathSciNet  CrossRef  MATH  Google Scholar 

  5. Shapiro H.S. Monotonic singular functions of high smoothness.-Michigan Math.J. 1968, 15, 265–275.

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Kahane J.-P. Trois notes sur les ensembles parfaits linéaires.-Enseignement Math. 1969, (2), 15, 185–192.

    MathSciNet  Google Scholar 

References

  1. Anderson J.M., Clunie J., Pommerenke Ch. On Bloch functions and normal functions.-J.Reine Angew. Math., 1974, 270, 12–37.

    MathSciNet  MATH  Google Scholar 

  2. Pommerenke Ch. Schlichte Funktionen und analytische Funktionen von beschränkter mittlerer Oszillation.-Comment.Math. Helv., 1977, 52, 591–602.

    MathSciNet  CrossRef  MATH  Google Scholar 

  3. Anderson J.M. On division by Inner Factors.-Comment. Math.Helv., 1979, 54, N 2, 309–317.

    MathSciNet  CrossRef  MATH  Google Scholar 

  4. Хавин В.П. О факториэации аналитических функций, гладких вплоть до границы.-Зан.научн.семин.ЛОМИ, 1971, 22, 202–205.

    Google Scholar 

  5. Гурарий В.П. О факториэации абсолютно сходяшихся рядов Тзйлора и интегралов Фурье.-Зап.научн.семин.ЛОМИ, 1972, 30, 15–32.

    Google Scholar 

References

  1. Chang S.-Y. A., Marshall D. A sharp inequality concerning the Dirichlet integral. 1982, preprint.

    Google Scholar 

  2. Moser J. A sharp form of an inequality by N.Trudinger.-Ind. Univ.Math.J., 1971, 20, 1077–1092.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Sarason D. Algebras between L∞ and H∞.-Lect.Notes in Math.Springer-Verlag, 1976, 512, 117–129.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Chang S.-Y. A. Structure of some subalgebra of L∞ of the torus.-Proc.Symposia in Pure Math., 1979, 35, Part 1, 421–426.

    MathSciNet  CrossRef  Google Scholar 

References

  1. Ahern P. The mean modulus and the derivative of an inner function.-Indiana Univ.Math.J., 1979, 28, 2, 311–347.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Uchiyama A. A constructive proof of the Fefferman — Stein decomposition of BMO (ℝn).-Acta Math.,1982, 148, 215–241.

    MathSciNet  CrossRef  MATH  Google Scholar 

  2. Uchiyama A. The Fefferman — Stein decomposition of smooth functions and its application to HP(ℝn). — University of Chicago, Ph.D.thesis, 1982.

    Google Scholar 

  3. Calderón A.P., Zygmund A. On higher gradients of harmonic functions.-Studia Math.,1964, 24, 211–226.

    MathSciNet  MATH  Google Scholar 

Reference

  1. Fefferman C., Stein E.M. Hp spaces of several variables.-Acta Math., 1972, 129, 137–193.

    MathSciNet  CrossRef  MATH  Google Scholar 

References

  1. Hoffman K. Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs, N.J., 1962.

    MATH  Google Scholar 

  2. Helson H. Lectures on Invariant Subspaces. Academic Press, New York, 1964.

    MATH  Google Scholar 

  3. Gamelin T. Uniform Algebras, Pretice-Hall. Englewood Cliffs, N.J., 1969.

    MATH  Google Scholar 

  4. Parreau M. Théorème de Fatou et problème de Dirichlet pour les lignes de Green de certaines surfaces de Riemann. — Ann.Acad.Sci.Fenn.Ser.A. I, 1958, no.250/25, 8 pp.

    Google Scholar 

  5. Widom H. Hp sections of vector bundles over Riemann surfaces.-Ann. of Math., 1971, 94, 304–324.

    MathSciNet  CrossRef  MATH  Google Scholar 

  6. Hasumi M. Invariant subspaces on open Riemann surfaces.-Ann.Inst.Fourier, Grenoble,1974, 24, 4, 241–286; II, ibid. 1976, 26, 2, 273–299.

    MathSciNet  CrossRef  MATH  Google Scholar 

  7. Brelot M. Topology of R.S. Martin and Green lines. Lectures on Functions of a Complex Variable, pp.105–121. Univ. of Michigan Press, Ann Arbor, 1955.

    Google Scholar 

  8. Hayashi M. Invariant subspaces on Riemann surfaces of Parreau-Widom type. Preprint (1980).

    Google Scholar 

  9. Voichick M. Extreme points of bounded analytic functions on infinitely connected regions.-Proc.Amer.Math.Soc., 1966, 17, 1366–1369.

    MathSciNet  MATH  Google Scholar 

  10. Neville C. Invariant subspaces of Hardy classes on infinitely connected open surfaces.-Memoirs of the Amer.Math.Soc., 1975, N 160.

    Google Scholar 

  11. Pommerenke Ch. On the Green's function of Fuchsian groups.-Ann.Acad.Sci.Fenn. Ser. A. I, 1976, 2, 408–427.

    MathSciNet  MATH  Google Scholar 

  12. Stanton C. Bounded analytic functions on a class of open Riemann surfaces.-Pacific J.Math., 1975, 59, 557–565.

    MathSciNet  CrossRef  MATH  Google Scholar 

  13. Pranger W. Riemann surfaces and bounded holomorphic functions.-Trans.Amer.Math.Soc., 1980, 259, 393–400.

    MathSciNet  CrossRef  MATH  Google Scholar 

Reference

  1. Casazza P.G.,Pengra R. and Sundberg C. Complemented ideals in the Disk Algebra.Israel J.Math.,vol.37.No1–2, (1980), p.76–83.

    MathSciNet  CrossRef  MATH  Google Scholar 

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Author information

Author notes

  1. Peter W. Jones

    Present address: Dept. of Mathematics, University of Chicago, 60637, Chicago, Illinois, USA

  2. Peter W. Jones

    Present address: Dept. of Mathematics, University of Chicago, 60637, Chicago, Illinois, USA

Authors and Affiliations

  1. Department of Mathematics, Washington University, Box 1146, 63130, St.Louis, MO, USA

    R. R. Coifman

  2. Faculté des Sciences d'Orsay, Université de Paris-Sud, France

    Yves Meyer

  3. Math. Dept., Rutgers University, 08903, New Brunswick, N.J., USA

    Benjamin Muckenhoupt

  4. Institut Mittag-Leffler, Auravãgen 17, S-182 62, Djursholm, Sweden

    Peter W. Jones

  5. Dept. of Mathematics, Yale University, 06520, New Haven, Connecticut, USA

    Stephen Semmes

  6. Washington University, Box 1146, 63130, St.Louis, MO, USA

    Richard Rochberg

  7. University of California, 90024, Los Angeles, California, USA

    John Garnett

  8. Washington University, 63130, St.Louis, Missouri, USA

    Albert Baernstein

  9. Dept.Math., University of California, 94720, Berkeley, California, USA

    Donald Sarason

  10. Department of Mathematics, University College, London, WC1E 6BT, London, England

    J. M. Anderson

  11. Math.Dept., University of Maryland, 20742, College Park, Maryland, USA

    Sun-Yung A. Chang

  12. Math.Dept., University of Maryland, 20742, College Park, Maryland, USA

    Sun-Yung A. Chang

  13. University of Wisconsin, Madison, Wisconsin, USA

    Patrick Ahern

  14. Institut Mittag-Leffler, Auravägen 17, S-182 62, Djursholm, Sweden

    Peter W. Jones

  15. Department of Mathematics, Ibaraki University, 310, Mito, Ibaraki, Japan

    Morisuke Hasumi

  16. Department of Mathematics, University of Missouri-Columbia, 65211, Columbia, Missouri, USA

    Peter G. Casazza

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    Coifman, R.R. et al. (1984). Singular integrals, BMO, Hp . In: Havin, V.P., Hruščëv, S.V., Nikol'skii, N.K. (eds) Linear and Complex Analysis Problem Book. Lecture Notes in Mathematics, vol 1043. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0072189

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