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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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The $H^ p$-corona theorem for the polydisc
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by Kai-Ching Lin
Trans. Amer. Math. Soc. 341 (1994), 371-375
DOI: https://doi.org/10.1090/S0002-9947-1994-1161426-6

Abstract:

Let ${H^p} = {H^p}({D^n})$ denote the usual Hardy spaces on the polydisc ${D^n}$. We prove in this paper the following theorem: Suppose ${f_1},{f_2}, \ldots ,{f_n} \in {H^\infty },{\left \| {{f_j}} \right \|_{{H^\infty }}} \leq 1$, and $\sum \nolimits _{j = 1}^m {|{f_j}(z)|} \geq \delta > 0$. Then for every $g$ in ${H^p}$, $1 < p < \infty$, there are ${H^p}$ functions $g,g, \ldots ,{g_m}$ such that $\sum \nolimits _{j = 1}^m {{f_j}(z){g_j}(z) = g(z)}$. Moreover, we have ${\left \| {{g_j}} \right \|_{{H^p}}} \leq c(m,n,\delta ,p){\left \| g \right \|_{{H^p}}}$. (When $p = 2,n = 1$, this theorem is known to be equivalent to Carleson’s corona theorem.)
References
  • É. Amar, On the corona problem, J. Geom. Anal. 1 (1991), no. 4, 291–305. MR 1129344, DOI 10.1007/BF02921307
  • M. Anderssonn, The ${H^2}$-corona theorem and ${\bar \partial _b}$, preprint.
  • Lennart Carleson, The corona theorem, Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968) Lecture Notes in Mathematics, Vol. 118, Springer, Berlin, 1970, pp. 121–132. MR 0264100
  • Sun-Yung A. Chang, Two remarks about $H^{1}$ and BMO on the bidisc, Conference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981) Wadsworth Math. Ser., Wadsworth, Belmont, CA, 1983, pp. 373–393. MR 730080
  • John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
  • John B. Garnett and Peter W. Jones, The corona theorem for Denjoy domains, Acta Math. 155 (1985), no. 1-2, 27–40. MR 793236, DOI 10.1007/BF02392536
  • T. W. Gamelin, Uniform algebras and Jensen measures, London Mathematical Society Lecture Note Series, vol. 32, Cambridge University Press, Cambridge-New York, 1978. MR 521440
  • G. M. Henkin, H. Lewy’s equation and analysis on pseudoconvex manifolds, Uspehi Mat. Nauk 32 (1977), no. 3(195), 57–118, 247 (Russian). MR 0454067
  • Peter W. Jones and Donald E. Marshall, Critical points of Green’s function, harmonic measure, and the corona problem, Ark. Mat. 23 (1985), no. 2, 281–314. MR 827347, DOI 10.1007/BF02384430
  • Kai-Ching Lin, $H^p$-solutions for the corona problem on the polydisc in $\textbf {C}^n$, Bull. Sci. Math. (2) 110 (1986), no. 1, 69–84 (English, with French summary). MR 861670
  • Nessim Sibony, Prolongement des fonctions holomorphes bornées et métrique de Carathéodory, Invent. Math. 29 (1975), no. 3, 205–230 (French). MR 385164, DOI 10.1007/BF01389850
  • Nessim Sibony, Problème de la couronne pour des domaines pseudoconvexes à bord lisse, Ann. of Math. (2) 126 (1987), no. 3, 675–682 (French). MR 916722, DOI 10.2307/1971364
  • N. Th. Varopoulos, BMO functions and the $\overline \partial$-equation, Pacific J. Math. 71 (1977), no. 1, 221–273. MR 508035
  • Nicholas Th. Varopoulos, Probabilistic approach to some problems in complex analysis, Bull. Sci. Math. (2) 105 (1981), no. 2, 181–224 (English, with French summary). MR 618877
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Bibliographic Information
  • © Copyright 1994 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 341 (1994), 371-375
  • MSC: Primary 46J15; Secondary 32A35
  • DOI: https://doi.org/10.1090/S0002-9947-1994-1161426-6
  • MathSciNet review: 1161426