Universal divisors in Hardy spaces

E. Amar; C. Menini

Studia Mathematica (2000)

  • Volume: 143, Issue: 1, page 1-21
  • ISSN: 0039-3223

Abstract

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We study a division problem in the Hardy classes H p ( ) of the unit ball of ℂ 2 which generalizes the H p corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a ℂ k -valued bounded Mholomorphic function B, with B | S = 0 , in order that for 1 ≤ p < ∞ and any function f ∈ H p ( ) with f | S = 0 there is a ℂ k -valued H p ( ) holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class H p ( ) is the entire module M S : = f ∈ H p ( ) : f | S = 0 . As a special case, for S = ∅, we get the H p corona theorem.

How to cite

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Amar, E., and Menini, C.. "Universal divisors in Hardy spaces." Studia Mathematica 143.1 (2000): 1-21. <http://eudml.org/doc/216806>.

@article{Amar2000,
abstract = {We study a division problem in the Hardy classes $H^\{p\}()$ of the unit ball of $ℂ^\{2\}$ which generalizes the $H^\{p\}$ corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a $ℂ^\{k\}$-valued bounded Mholomorphic function B, with $B_\{|S\} = 0$, in order that for 1 ≤ p < ∞ and any function $f ∈ H^\{p\}()$ with $f_\{|S\} = 0$ there is a $ℂ^\{k\}$-valued $H^\{p\}()$ holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class $H^\{p\}()$ is the entire module $M_\{S\}:= \{f ∈ H^\{p\}(): f_\{|S\} = 0 \}$. As a special case, for S = ∅, we get the $H^\{p\}$ corona theorem.},
author = {Amar, E., Menini, C.},
journal = {Studia Mathematica},
keywords = {Hardy space; unit ball; divisors; corona problem},
language = {eng},
number = {1},
pages = {1-21},
title = {Universal divisors in Hardy spaces},
url = {http://eudml.org/doc/216806},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Amar, E.
AU - Menini, C.
TI - Universal divisors in Hardy spaces
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 1
SP - 1
EP - 21
AB - We study a division problem in the Hardy classes $H^{p}()$ of the unit ball of $ℂ^{2}$ which generalizes the $H^{p}$ corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a $ℂ^{k}$-valued bounded Mholomorphic function B, with $B_{|S} = 0$, in order that for 1 ≤ p < ∞ and any function $f ∈ H^{p}()$ with $f_{|S} = 0$ there is a $ℂ^{k}$-valued $H^{p}()$ holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class $H^{p}()$ is the entire module $M_{S}:= {f ∈ H^{p}(): f_{|S} = 0 }$. As a special case, for S = ∅, we get the $H^{p}$ corona theorem.
LA - eng
KW - Hardy space; unit ball; divisors; corona problem
UR - http://eudml.org/doc/216806
ER -

References

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  7. [7] L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943-949. Zbl0172.41701
  8. [8] C. Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), 201-213. 
  9. [9] W. Rudin, Function Theory in the Unit Ball of ℂ n , Grundlehren Math. Wiss. 241, Springer,1980. 
  10. [10] H. Skoda, Valeurs au bord pour les solutions de l'équation d'', et caractérisation des zéros des fonctions de la classe de Nevanlinna, Bull. Soc. Math. France 104 (1976), 225-299. Zbl0351.31007
  11. [11] N. Varopoulos, BMO functions and the ∂ ¯ -equation, Pacific J. Math. 71 (1977), 221-273. 

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