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The $ H\sp p$-corona theorem for the polydisc


Author: Kai-Ching Lin
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
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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\Vert {{f_j}} \right\Vert _{{H^\infty }}} \leq 1$, and $ \sum\nolimits_{j = 1}^m {\vert{f_j}(z)\vert} \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\Vert {{g_j}} \right\Vert _{{H^p}}} \leq c(m,n,\delta ,p){\left\Vert g \right\Vert _{{H^p}}}$. (When $ p = 2,n = 1$, this theorem is known to be equivalent to Carleson's corona theorem.)


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1994-1161426-6
Keywords: Corona theorem, Carleson measures, polydisc
Article copyright: © Copyright 1994 American Mathematical Society

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