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Capacities in Complex Analysis pp 116–147Cite as

Szegö Kernels

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Part of the book series: Aspects of Mathematics / Aspekte der Mathematik ((ASMA,volume E 14))

Abstract

We keep the notation from Section X.

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Notes and references

  1. General references for this section are Bungart [3] and Rudin [7].

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  2. Theorem XI: 3 was proved by a combination of arguments of Glicksberg, König, Seever and Rainwater. See Rudin [7, pg. 194] for details.

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  3. Theorem XI: 5 is due to Henkin, see Henkin and Čirka [4].

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  4. We have no exact description of the sets that satisfies condition (**), (pg 94).

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  5. By taking dilatations, it is clear that every starshaped domain satisfies (**). It is known that this is true also for smooth strictly pseudoconvex domains, cf.: N. Kerzman; Hölder and Lp estimates for solutions of \(\bar \partial u = f\) in strongly pseudoconvex domains, Comm. Pure Appl. Math. 24 (1971), 301–379

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  6. Brian Cole and R. Michael Range; A-measures on complex manifolds and some applications, J. Func. An. 11 (1972), 393–400.

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  7. That H∞+C is a closed subalgebra of L∞ was first proved by Sarason [8] on the unit circle, by Rudin [6] on the unit sphere in ₵n, by Aytuna and Chollet [1] on the boundary of strictly pseudoconvex domains.

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  8. Jewell and Krantz [5] proved the results for convex sets in ₵n with real analytic boundary and they also remarked that they had the result for ∂Dα:⊂₵n where H+C. The methods in the above papers are different from ours.

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  9. Aytuna, A and Chollet, A.-M., Une Extension d’un resultat de W. Rudin. Bull. Soc. Math. france 104 (1976), 383–388.

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  10. Bonami, A. and Lohoué, N., Projecteurs de Bergman et Szegö pour une classe de domains faiblement pseudo-convexes et estimations Lp. Compos. Math. 46 (1982), 159–226.

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  11. Bungart, L., Boundary kernel functions for domains on complex manifolds. Pacific J. Math 14 (1964), 1151–1164.

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  12. Henkin, G.M. and Čirka, E.M., Boundary properties of holomorphic functions of several complex variables. J. Soviet Math. 5 (1976), 612–687.

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  13. Jewell, N.P. and Krantz, S.G., Toeplitz operators and related function algebras on certain pseudoconvex domains. Trans. Amer. Math. Soc. 252 (1979), 297–312.

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  14. Rudin, W., Spaces of type H∞+C. Ann. Inst. Fourier 25 (1975), 99–125.

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  15. Rudin, W., Function theory in the unit ball of ₵n. Springer-Verlag. New York, Heidelberg, Berlin 1980.

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  16. Sarason, D., Generalized interpolation in H∞. Trans. Amer. Math. Soc. 127 (1967), 179–203.

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Authors and Affiliations

  1. Department of Mathematics, University of Umeå, Sweden

    Prof. Dr. Urban Cegrell

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  1. Prof. Dr. Urban Cegrell
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© 1988 Springer Fachmedien Wiesbaden

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Cegrell, U. (1988). Szegö Kernels. In: Capacities in Complex Analysis. Aspects of Mathematics / Aspekte der Mathematik, vol E 14. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-14203-4_12

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  • DOI: https://doi.org/10.1007/978-3-663-14203-4_12

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-528-06335-1

  • Online ISBN: 978-3-663-14203-4

  • eBook Packages: Springer Book Archive

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