Skip to main content

Function Theory in the Ball

  • Chapter
Several Complex Variables II

Part of the book series: Encyclopaedia of Mathematical Sciences ((EMS,volume 8))

Abstract

In the theory of functions of several complex variables, the ball occupies a crucial position: on the one hand, the ball is the simplest example of a strictly pseudoconvex domain with smooth boundary; on the other hand, it is the simplest bounded classical domain.

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 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight 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.

References

  • Ahern, P.R., Schneider, R. (1980): A smoothing property of the Henkin and Szegö projections. Duke Math. J. • •, No. 1, 135–143, Zbl.453.32004.

    Google Scholar 

  • Aizenberg, L.A., Yuzhakov, A.P. (1979): Integral representations and residues in multidimensional complex analysis. Novosibirsk: Nauka. 335 pp. Engl. transl.: Transl. Math. Monogr., Vol. 58, Providence, 283 pp. (1983), Zbl.445.32002.

    Google Scholar 

  • Aleksandrov, A.B. (1981): Essays on non locally convex Hardy classes. Lect. Notes Math. 864, 1–89. Berlin, Heidelberg, New York: Springer-Verlag, Zbl.482.46035.

    Google Scholar 

  • Aleksandrov, A.B. (1982): Existence of inner functions in the ball. Mat. Sb., Nov. Ser. 118, No. 2, 147–163. Engl. Transl.: Math. USSR, Sb. 46, 143-159 (1983), Zbl.503.32001.

    Google Scholar 

  • Aleksandrov, A.B. (1983): On the boundary values of functions holomorphic in the ball. Dokl. Akad. Nauk SSSR 274, No. 4, 777–779. Engl. Transl.: Sov. Math., Dokl. 28, 134-137 (1983), Zbl.543.32002.

    Google Scholar 

  • Aleksandrov, A.B. (1984): Inner functions on compact spaces. Funkts. Anal. Prilozh. 18, No. 2, 1–13. Engl. transl.: Funct. Anal. Appl. 18, 87-98 (1984), Zbl.574.32006.

    Google Scholar 

  • Alexander, H. (1982): On zero sets for the ball algebra. Proc. Am. Math. Soc. 86, No. 1, 71–74, Zbl.504.32005.

    MATH  Google Scholar 

  • Amar, E. (1982): Sur le volume des zéros des fonctions holomorphes et bornées dans la boule de ℂn. Proc. Am. Math. Soc. 85, No. 1, 47-52, Zbl.507.32001.

    Google Scholar 

  • Amar, E. (1983a): Non division dans A (Ω). C. R. Acad. Sci., Sér. 1, 296, No. 13, 541–544. See also: Math. Z. 188, 493-511 (1985), Zbl.547.32010.

    MathSciNet  Google Scholar 

  • Amar, E. (1983b): Extension de fonctions holomorphes et courants. Bull. Sci. Math., II. Ser. 107, 25–48, Zbl.543.32007.

    MathSciNet  MATH  Google Scholar 

  • Axler, Sh., Shapiro, J.H. (1983): Putnam’s theorem, Alexander’s spectral area estimate, and VMO. Math. Ann. 271, 161–183 (1985), Zbl.541.30021.

    MathSciNet  Google Scholar 

  • Bartolomeis, P. de, Tomassini, G. (1981): Idéaux de type fini dans A (D). C. R. Acad. Sci., Paris, Sér. I, 293, No 2, 133–134, Zbl.477.32016.

    MATH  Google Scholar 

  • Berndtsson, B. (1980): Integral formulas for the ∂∂-equation and zeros of bounded holomorphic functions in the unit ball. Math. Ann. 249, No. 2, 163–176, Zbl.414.31007.

    MathSciNet  MATH  Google Scholar 

  • Bishop, E. (1965): Difierentiable manifolds in complex Euclidean space. Duke Math. J. 32, No. 1, 1–21, Zbl.154, 85.

    MathSciNet  MATH  Google Scholar 

  • Boas, R. (1955): Isomorphism between H p and L p. Am. J. Math. 77, No. 4, 655–656, Zbl.65, 345.

    MathSciNet  MATH  Google Scholar 

  • Bourgain, J. (1982): The non-isomorphism ofH 1spaces in one and several variables. J. Funct. Anal. 46, No. 1, 45–57, Zbl.492.46043.

    MathSciNet  MATH  Google Scholar 

  • Bourgain, J. (1983a): The non-isomorphism of H 1-spaces in a different number of variables. Bull. Soc. Math. Belg., Ser. B 35, No. 2, 127–136, Zbl.533.46036.

    MathSciNet  MATH  Google Scholar 

  • Bourgain, J. (1983b): The dimension conjecture for polydisc algebras. Isr. J. Math. 48, 289–304 (1984), Zbl.572.46047.

    MathSciNet  Google Scholar 

  • Chaumat, J., Chollet, A.-M. (1979): Ensembles pics pour A (D). Ann. Inst. Fourier 29, No. 3, 171–200, Zbl.398.32004.

    MathSciNet  MATH  Google Scholar 

  • Chaumat, J., Chollet, A.-M. (1980): Caractérisation et propriétés des ensembles localement pics de A (D). Duke Math. J. 47, No. 4, 763–787, Zbl.454.32013.

    MathSciNet  MATH  Google Scholar 

  • Chaumat, J., Chollet, A.-M. (1982): Ensembles pics pour A (D) non globalement inclus dans une variété intégrale. Math. Ann. 258, No. 3, 243–252, Zbl.574.32023.

    MathSciNet  MATH  Google Scholar 

  • Chaumat, J., Chollet, A.-M. (1983): Ensembles des zéros et d’interpolation le long de courbes. C. R. Acad. Sci., Paris, Sér. I 296, No. 19, 789–792, Zbl.566.32010.

    MathSciNet  MATH  Google Scholar 

  • Chirka, E.M. (1973): The theorems of Lindelöf and Fatou in ℂn. Mat. Sb., Nov. Ser. 92, No. 4 622–644. Engl. transl.: Math. USSR, Sb. 21, 619-639 (1975), Zbl.297.32001.

    Google Scholar 

  • Chirka, E.M., Khenkin, G.M. Henkin, G.M. (1975): Boundary properties of holomorphic functions of several complex variables. In Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 4, 13–142. Engl. transl.: J. Sov. Math. 5, 612-687 (1976), Zbl.375.32005.

    Google Scholar 

  • Coifman, R.R. (1974): A real variable characterization of H p. Stud. Math. 51, No. 3, 269–274, Zbl.289.46037.

    MathSciNet  MATH  Google Scholar 

  • Coifman, R.R., Rochberg, R. (1980): Representation theorems for holomorphic and harmonic functons. Astérisque 77, 12–66, Zbl.472.46040.

    Google Scholar 

  • Coifman, R.R., Rochberg, R., Weiss, G. (1976): Factorization theorems for Hardy spaces in several variables. Ann. Math., II, Ser. 103, No. 3, 611–635, Zbl.326.32011.

    MathSciNet  MATH  Google Scholar 

  • Coifman, R.R., Weiss, G. (1977): Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, No. 4, 569–645, Zbl.358.30023.

    MathSciNet  MATH  Google Scholar 

  • Cumenge, A. (1983): Extension dans des classes de Hardy de fonctions holomorphes et estimations de type “mesure de Carleson” pour l’équation ∂. Ann. Inst. Fourier 33, No. 3, 59–97, Zbl.487.32011.

    MathSciNet  MATH  Google Scholar 

  • Dautov, Sh.A, Khenkin, G.M. Henkin, G.M. (1978): Zeros of holomorphic functions of finite order and weighted estimates for solutions of the ∂;-equation. Mat. Sb. 107, No. 2, 163–174. Engl. transl.: Math. USSR, Sb. 35, 449-459 (1979), Zbl.392.32001.

    MathSciNet  Google Scholar 

  • Davie, A.M., Jewell, N.P. (1977): Toeplitz operators in several complex variables. J. Funct. Anal. 26, No. 4, 356–368, Zbl.374.47011.

    MathSciNet  MATH  Google Scholar 

  • Davie, A., Öksendal, B. (1972): Peak interpolation sets for some algebras of analytic functions. Pac. J. Math. 41, No. 1, 81–87, Zbl.218, 292.

    Google Scholar 

  • Duchamp, Th., Stout, E.L. (1981): Maximum modulus sets. Ann. Inst. Fourier 31, No. 3, 37–69, Zbl.439.32007.

    MathSciNet  MATH  Google Scholar 

  • Duren, P.L. (1970): Theory of H p Spaces. 258 pp. New York: Academic Press, Zbl.215, 202.

    MATH  Google Scholar 

  • Duren, P.L., Romberg, B.W., Shields, A.L. (1969): Linear functionals on H p spaces with 0 < p < 1. J. Reine Angew. Math. 238, No. 1, 32–60, Zbl. 176, 431.

    MathSciNet  MATH  Google Scholar 

  • Fefferman, C., Stein, E.M. (1972): H p spaces of several variables. Acta Math. 129, No. 3-4, 137–193, Zbl.257.46078.

    MathSciNet  MATH  Google Scholar 

  • Forelli, F. (1974): Measures whose Poisson integrals are pluriharmonic. III. J. Math. 18, No. 3, 373–388, Zbl.296.31014.

    MathSciNet  MATH  Google Scholar 

  • Forelli, F. (1975): Measures whose Poisson integrals are pluriharmonic II. III. J. Math. 19, No. 4, 584–592, Zbl.329.31002.

    MathSciNet  MATH  Google Scholar 

  • Forelli, F. (1977): A necessary condition on the extreme points of a class of holomorphic functions. Pac. J. Math. 73, No. 1, 81–86, Zbl.346.32002.

    MathSciNet  MATH  Google Scholar 

  • Forelli, F. (1979): Some extreme rays of the positive pluriharmonic functions. Can. J. Math. 31, No. 1, 9–16, Zbl.373.31005.

    MathSciNet  MATH  Google Scholar 

  • Forelli, F., Rudin, W. (1974): Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24, No. 6, 593–602, Zbl.297.47041.

    MathSciNet  MATH  Google Scholar 

  • Fornaess, J.E., Henriksen, B.S. (1982): Characterization of global peak sets for A (D). Math. Ann. 259, No. 1, 125–130, Zbl.489.32010.

    MathSciNet  MATH  Google Scholar 

  • Garnett, J.B. (1981): Bounded Analytic Functions. 467 pp. New York, London, Toronto, Sydney, San Francisco: Academic Press, Zbl.469.30024.

    MATH  Google Scholar 

  • Garnett, J.B., Latter, R.H. (1978): The atomic decomposition for Hardy spaces in several complex variables. Duke Math. J. 45, No. 4, 815–845, Zbl.403.32006.

    MathSciNet  MATH  Google Scholar 

  • Gokhberg, I.Ts., Krupnik, N.Ya. (1973): Introduction to the Theory of One-Dimensional Singular Integral Operators. 426 pp. Kishinev: Shtiintsa. German transl.: Basel, Boston, Stuttgart: Birkhäuser 1979, Zbl.271.47017.

    Google Scholar 

  • Gowda, M.S. (1983): Nonfactorization theorems in weighted Bergman and Hardy spaces on the unit ball of ℂn (n > 1). Trans. Am. Math. Soc. 277, No. 1, 203–212, Zbl.526.32005.

    MathSciNet  MATH  Google Scholar 

  • Greiner, P.C., Stein, E.M. (1977): Estimates for the ∂-Neumann problem. Math. Notes 19. Princeton, N.J.: Princeton Univ. Press, 194 pp., Zbl.354.35002.

    Google Scholar 

  • Hakim, M. (1983): Valeurs au bord de fonctions holomorphes bornées en plusieurs variables complexes. Sémin. Bourbaki, 35e année, Vol. 1982/83, Exp. No. 613, Astérisque 105–106, 293-305, Zbl.519.32007.

    Google Scholar 

  • Hakim, M., Sibony, N. (1978): Ensembles pics dans des domaines strictement pseudoconvexes. Duke Math. J. 45, No. 3, 601–617, Zbl.402.32008.

    MathSciNet  MATH  Google Scholar 

  • Hakim, M., Sibony, N. (1982a): Ensemble des zéros d’une fonction holomorphe bornée dans la boule unité. Math. Ann. 260, No. 4, 469–474, Zbl.499.32006.

    MathSciNet  MATH  Google Scholar 

  • Hakim, M., Sibony, N. (1982b): Fonctions holomorphes bornées sur la boule unité de ℂn. Invent. Math. 67, No. 2, 213–222, Zbl.475.32007.

    MathSciNet  MATH  Google Scholar 

  • Hakim, M., Sibony, N. (1983): Valeurs au bord des modules de fonctions holomorphes. Math. Ann. 264, No. 2, 197–210, Zbl.514.32008.

    MathSciNet  MATH  Google Scholar 

  • Henriksen, B.S. (1982): A peak set of Hausdorff dimension 2n — 1 for the algebra A(D) in the boundary of a domain D with C -boundary in ℂn. Math. Ann. 259, No. 2, 271–277, Zbl.483.32011.

    MathSciNet  MATH  Google Scholar 

  • Hoffman, K. (1962): Banach Spaces of Analytic Functions. Englewood Cliffs, N.J.: Prentice Hall, Inc., 217 pp., Zbl. 117, 340.

    MATH  Google Scholar 

  • Hörmander, L. (1967): L p estimates for (pluri-) subharmonic functions. Math. Scand. 20, No. 1, 65–78, Zbl.156, 122.

    MathSciNet  MATH  Google Scholar 

  • Horowitz, C. (1977): Factorization theorems for functions in the Bergman spaces. Duke Math. J. 44, No. 1, 201–213, Zbl.362.30031.

    MathSciNet  MATH  Google Scholar 

  • Hurewicz, W., Wallman, H. (1941): Dimension Theory. Princeton: Princeton University Press, 165 pp., Zbl.60, 398.

    Google Scholar 

  • Jakóbczak, P. (1983): On the regularity of extension to strictly pseudoconvex domains of functions holomorphic in a submanifold in general position. Ann. Pol. Math. 42, 115–124, Zbl.552.32009.

    MATH  Google Scholar 

  • Janson, S. (1976): On functions with conditions on the mean oscillation. Ark. Mat. 14, No. 2, 189–196, Zbl.341.43005.

    MathSciNet  MATH  Google Scholar 

  • Jewell, N.P. (1980): Toeplitz operators on the Bergman spaces and in several complex variables. Proc. Lond. Math. Soc., III. Ser. 41, No. 2, 193–216, Zbl.412.47014.

    MathSciNet  MATH  Google Scholar 

  • Jöricke, B. (1981): On pseudoanalytic continuation and the behavior of the boundary values of analytic functions on small sets. Preprint, Akad. Wiss. DDR, Inst. Math. P-Math 22/81, 24 pp., Zbl.466.30005.

    Google Scholar 

  • Jöricke, B. (1982): The two-constants Theorem for functions of several complex variables. Math. Nachr. 107, 17–52 (Russian), Zbl.526.32003.

    MathSciNet  MATH  Google Scholar 

  • Khenkin, G.M. Henkin, G.M. (1968): The Banach space of analytic functions in the sphere and in the bicylinder are not isomorphic. Funkts. Anal. Prilozh. 2, No. 4, 82–91. Engl. transl: Funct. Anal. Appl. 2, 334-341 (1968), Zbl.181, 134.

    Google Scholar 

  • Khenkin, G.M. Henkin, G.M. (1971): Approximation of functions in pseudoconvex domains and a theorem of Lejbenzon. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 19, No. 1, 37–42 (Russian), Zbl.214, 337.

    Google Scholar 

  • Khenkin, G.M. Henkin, G.M. (1972): Continuation of bounded holomorphic functions from submanifolds in general ppsition to strictly pseudoconvex domains. Izv. Akad. Nauk SSSR, Ser. Mat. 36, No. 3, 540–567. Engl. transl.: Math. USSR, Izv. 6, 536-563 (1973), Zbl.249.32009.

    MathSciNet  Google Scholar 

  • Khenkin, G.M. Henkin, G.M. (1977a): The equation of H. Lewy and analysis on pseudoconvex manifolds I. Usp. Mat. Nauk 32, No. 3, 57–118. English transl: Russ. Math. Surv. 32, No. 3, 59-130 (1977), Zbl.358.35057.

    MATH  Google Scholar 

  • Khenkin, G.M. Henkin, G.M. (1977b): The equation of H. Lewy and analysis on pseudoconvex manifolds II. Mat. Sb., Nov. Ser. 102, No. 1, 71–108. Engl. transl.: Math. USSR, Sb. 31, 63-94 (1977), Zbl.388.35052.

    MathSciNet  Google Scholar 

  • Khenkin, G.M. Henkin, G.M., Leiterer, J. (1984): Theory of Functions on Complex Manifolds. 226 pp. Berlin: Akademie-Verlag, Zbl.573.32001.

    Google Scholar 

  • Khenkin, G.M. Henkin, G.M., Mityagin, B.S. (1971): Linear problems of complex analysis. Usp. Mat. Nauk 26, No. 4, 93–152. Engl. transl.: Russ. Math. Surv. 26 (1971), No. 4, 99-164 (1972), Zbl.245.46027.

    Google Scholar 

  • Khenkin, G.M. Henkin, G.M., Tumanov, A.E. (1976): Interpolation submanifolds of pseudoconvex manifolds. Math. program. rel. Probl., Cent. Ehkon. Mat. Inst. Akad. Nauk SSSR, Mosk. 1974, 74–86. Engl. transl.: transl., II. Ser, Am. Math. Soc. 115, 59-69 (1980), Zbl.455.32009.

    Google Scholar 

  • Khrushchev, S.V., Peller, V.V. (1982): Hankel operators, best approximation and stationary Gaussian processes. Usp. Mat. Nauk 37, No. 1, 53–124. Engl. transl.: Russ. Math. Surv. 37, No. 1, 61-144 (1982), Zbl.497.60033.

    MathSciNet  MATH  Google Scholar 

  • Khrushchev, S.V., Vinogradov, S.A. (1981): Free interpolation in the space of uniformly convergent Taylor series. Lect. Notes Math. 864, 171–213, Zbl.463.30001.

    Google Scholar 

  • Koosis, P. (1980): Introduction to H p-spaces. Lond. Math. Soc. Lect. Notes Ser. No. 40, 376 pp., London: Cambridge Univ. Press. Zbl.435.30001.

    Google Scholar 

  • KoraĔyi, A, Vagi, S. (1971): Singular integrals in homogeneous spaces and some problems classical analysis. Ann. Sc. Norm. Super. Pisa, Sci. Fij. Mat., III, Ser. 25, No. 4, 575–648, Zbl.291.43014.

    Google Scholar 

  • Korenblum, B. (1975): An extension of the Nevanlinna theory. Acta Math. 135, No. 3-4, 187–219, Zbl.323.30030.

    MathSciNet  MATH  Google Scholar 

  • Krantz, S.G. (1980): Holomorphic functions of bounded mean oscillation and mapping properties of the Szegö projection. Duke Math. J. 47, No. 4, 743–761, Zbl.456.32004.

    MathSciNet  MATH  Google Scholar 

  • Lindenstrauss, J, PelcziĔski, A. (1971): Contributions to the theory of the classical Banach spaces. J. Funct. Anal. 8, No. 2, 225–249, Zbl.224.46041.

    MATH  Google Scholar 

  • Löw, E. (1982): A construction of inner functions on the unit ball in ℂn;. Invent. Math. 67, No. 2, 223–229, Zbl.528.32006.

    MathSciNet  Google Scholar 

  • Löw, E. (1984): Inner functions and boundary values in H (Ω) and A(Ω) is smoothly bounded pseudoconvex domains. Math. Z. 185, No. 2, 191–210, Zbl.508.32005.

    MathSciNet  Google Scholar 

  • Lumer, G. (1971): Espaces de Hardy en plusieurs variables complexes. C. R. Acad. Sci, Paris, Sér. A 273, No. 3, 151–154, Zbl.216, 161.

    MathSciNet  MATH  Google Scholar 

  • Malliavin, P. (1974): Fonctions de Green d’un ouvert strictement pseudoconvex et classe de Nevanlinna. C. R. Acad. Sci, Paris, Sér. A 278, No. 3, 141–144, Zbl.279.32005.

    MathSciNet  MATH  Google Scholar 

  • McDonald, G. (1979): The maximal ideal space of H + C on the ball in ℂn. Can. J. Math. 31, No. 1, 79–86, Zbl.412.46045.

    MATH  Google Scholar 

  • Nagel, A, Wainger, S. (1981): Limits of bounded holomorphic functions along curves. Ann. Math. Stud. 100, 327–344, Zbl.517.32003.

    MathSciNet  Google Scholar 

  • Nikol’skij, N. (1986): Treatise on the Shift Operator. Spectral Function Theory. New York, Berlin, Heidelberg: Springer-Verlag, 505 pp. Enlarged English edition of the Russian original, Moscow: Nauka 1980, 304 pp., Zbl.587.47036.

    Google Scholar 

  • PelczyĔski, A. (1977): Banach spaces of analytic functions and absolutely summing operators. Amer. Math. Soc., Providence, R. I, 92 pp., Zbl.475.46022.

    Google Scholar 

  • Pinchuk, S.I. (1974): A boundary uniqueness theorem for holomorphic functions of several complex variables. Mat. Zametki 15, No. 2, 205–212. Engl. transl.: Math. Notes. 15, 116-120 (1974), Zbl.285.32002.

    MathSciNet  Google Scholar 

  • Pinchuk, S.I. (1986): Holomorphic mappings in ℂn and the problem of holomorphic equivalence. Itogi Nauki Tekh, Ser. Sovrem. Probi. Mat, Fundam. Napravleniya 9, 195–223. English transl.: Several Complex Variables III, Encycl. Math. Sci. 9, 173-199, Berlin, Heidelberg, New York: Springer-Verlag 1989, Zbl.658.32011.

    MathSciNet  Google Scholar 

  • Piranian, G. (1966): Two monotonic, singular, uniformly almost smooth functions. Duke Math. J. 33, No. 2, 255–262, Zbl.143, 74.

    MathSciNet  MATH  Google Scholar 

  • Power, S.C. (1985): Hörmander’s Carleson theorem for the ball. Glasg. Math. J. 26, No. 1, 13–17, Zbl.576.32007.

    MathSciNet  MATH  Google Scholar 

  • Rham, G. de (1955): Variétés Différentiables. Paris: Hermann, 196 pp., Zbl.65, 324.

    MATH  Google Scholar 

  • Rothschild, L.P., Stein, E.M. (1976): Hypoelliptic differential operators and nilpotent groups. Acta Math. 137, No. 3–4, 247–320, Zbl.346.35030.

    MathSciNet  Google Scholar 

  • Rudin, W. (1969): Function Theory in Polydiscs. New York: Benjamin, 188 pp., Zbl.177, 341.

    MATH  Google Scholar 

  • Rudin, W. (1980): Function Theory in the Unit Ball of ℂn. New York, Berlin, Heidelberg: Springer-Verlag, 436 pp., Zbl.495.32001.

    Google Scholar 

  • Rudin, W. (1983): Inner functions in the unit ball of ℂn. J. Funct. Anal. 50, No. 1, 100–126, Zbl.554.32002.

    MathSciNet  MATH  Google Scholar 

  • Ryll, J., Wojtaszsczyk, P. (1983): On homogeneous polynomials on a complex ball. Trans. Am. Math. Soc. 276, No. 1, 107–116, Zbl.522.32004.

    MATH  Google Scholar 

  • Sadullaev, A. (1976a): A boundary uniqueness theorem in ℂn. Mat. Sb., Nov. Ser. 101, No. 4, 568–583. Engl. transl.: Math. USSR, Sb. 30, 501-514 (1978), Zbl.346.32024.

    MathSciNet  Google Scholar 

  • Sadullaev, A. (1976b): On inner functions in ℂn. Mat. Zametki 19. No. 1, 63–66. Engl. transl.: Math. Notes 19, 37-38 (1976), Zbl.335.32002.

    MathSciNet  Google Scholar 

  • Shabat, B.V. (1976): Introduction to Complex Analysis, Part II. Moscow: Nauka. 400 pp. French transl.: Moscow: MIR 1990, Zbl.578.32001; Zbl.188, 379.

    Google Scholar 

  • Skoda, H. (1976): Valeurs au bord pour les solutions de l’opérateur d″ et caractérisation des zéros des fonctions de la classe de Nevanlinna. Bull. Soc. Math. Fr. 104, No. 3, 225–299, Zbl.351.31007.

    MathSciNet  MATH  Google Scholar 

  • Stein, E.M. (1970): Singular Integrals and Differentiability Properties of Functions. Princeton, N.J.: Princeton Univ. Press, 290 pp., Zbl.207, 135.

    MATH  Google Scholar 

  • Stein, E.M. (1973): Singular integrals and estimates for the Cauchy-Riemann equations. Bull. Am. Math. Soc. 79, No. 2, 440–445, Zbl.257.35040.

    MATH  Google Scholar 

  • Stout, E.L. (1982): The dimension of peak-interpolations sets. Proc. Am. Math. Soc. 86, No. 3, 413–416, Zbl.502.32012.

    MathSciNet  MATH  Google Scholar 

  • Szegö, G. (1959): Orthogonal Polynomials. Am. Math. Soc., Colloq. Publ. 23, 421 pp., Zbl.89, 275.

    Google Scholar 

  • Tamm, M. (1982): Sur l’image par une fonction holomorphe bornée du bord d’un domaine pseudoconvexe. C. R. Acad. Sci., Paris, Sér. I 294, No. 16, 537–540, Zbl.497.32003.

    MathSciNet  MATH  Google Scholar 

  • Tomaszewski, B. (1984): Interpolation and inner maps that preserve measure. J. Funct. Anal. 55, No. 1, 63–67, Zbl.531.32002.

    MathSciNet  MATH  Google Scholar 

  • Tumanov, A.E. (1977): A peak set of metric dimension 2.5 for the algebra of holomorphic functions on the three-dimensional sphere in ℂn. Izv. Akad. Nauk SSSR, Ser. Mat. 41, No. 2, 370–377. Engl. transl.: Math. USSR, Izv. 11, 353-359 (1977), Zbl.368.46048.

    MathSciNet  MATH  Google Scholar 

  • Val’skij, R.Eh. (1971): On measures orthogonal to analytic functions in ℂn. Dokl. Akad. Nauk SSSR 198, No. 3, 502–505. Engl. transl.: Sov. Math., Dokl. 12, 808-812 (1971), Zbl.234.46056.

    MathSciNet  Google Scholar 

  • Varopoulos, N.Th. (1980): Zeros of H p functions in several complex variables, Pac. J. Math. 88, No. 1, 189–246, Zbl.454.32006.

    MathSciNet  MATH  Google Scholar 

  • Wells, R.O. jun. (1973): Differential Analysis on Complex Manifolds. Englewood Cliffs, N.J.: Prentice-Hall, Inc., 252 pp. 2nd ed.: New York, Berlin, Heidelberg: Springer-Verlag 1980, Zbl.262.32005.

    MATH  Google Scholar 

  • Widom, H. (1964): On the spectrum of Toeplitz operators. Pac. J. Math. 14, No. 1, 365–375, Zbl. 197, 109.

    MathSciNet  MATH  Google Scholar 

  • Wojtaszczyk, P. (1982): On functions in the ball algebra. Proc. Am. Math. Soc. 85, No. 2, 184–186, Zbl.503.32005.

    MathSciNet  MATH  Google Scholar 

  • Wojtaszczyk, P. (1983): Hardy spaces on the complex ball are isomorphic to Hardy spaces on the disc, 1 ≤ p < +∞. Ann. Math., II. Ser. 118, No. 1, 21–34, Zbl.546.32003.

    MathSciNet  MATH  Google Scholar 

Download references

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1994 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Aleksandrov, A.B. (1994). Function Theory in the Ball. In: Khenkin, G.M., Vitushkin, A.G. (eds) Several Complex Variables II. Encyclopaedia of Mathematical Sciences, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-57882-3_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-57882-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-63391-1

  • Online ISBN: 978-3-642-57882-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics