Universal divisors in Hardy spaces

Amar, E.; Menini, C.

Amar, E. ; Menini, C.

Studia Mathematica, Tome 141 (2000), p. 1-21 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $f \in H^{p} ()$ 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 $M_{S} : = f \in H^{p} () : f_{| S} = 0$ . As a special case, for S = ∅, we get the $H^{p}$ corona theorem.

Publié le : 2000-01-01

Zbl 0967.32006

EUDML-ID : urn:eudml:doc:216806

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     author = {E. Amar and C. Menini},
     title = {Universal divisors in Hardy spaces},
     journal = {Studia Mathematica},
     volume = {141},
     year = {2000},
     pages = {1-21},
     zbl = {0967.32006},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv143i1p1bwm}
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Amar, E.; Menini, C. Universal divisors in Hardy spaces. Studia Mathematica, Tome 141 (2000) pp. 1-21. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv143i1p1bwm/

Bibliographie

[000] [1] D. & E. Amar, Sur les suites d'interpolation en plusieurs variables, Pacific J. Math. 75 (1978), 15-20. | Zbl 0392.32002

[001] [2] E. Amar, On the corona problem, J. Geom. Anal. 1 (1991), 291-305. | Zbl 0794.32007

[002] [3] E. Amar, Interpolating sequences for $H^{\infty} ()$ in the ball of $ℂ^{n}$ , Ark. Mat., to appear.

[003] [4] E. Amar et A. Bonami, Mesures de Carleson d’ordre α et solutions au bord de l’équation $\bar{\partial}$ , Bull. Soc. Math. France 107 (1979), 23-48. | Zbl 0409.46035

[004] [5] M. Andersson and H. Carlsson, Estimates of solutions of the $H^{p}$ and BMOA corona problem, Math. Ann. 316 (2000), 83-102.

[005] [6] G. Henkin, H. Lewy's equation and analysis on pseudoconvex manifolds, Part I, Russian Math. Surveys 32 (1977), no. 3, 59-130; Part II, Math. USSR-Sb. 31 (1977), no. 1, 63-94.

[006] [7] L. Hörmander, Generators for some rings of analytic functions, Bull. Amer. Math. Soc. 73 (1967), 943-949. | Zbl 0172.41701

[007] [8] C. Horowitz, Factorization theorems for functions in the Bergman spaces, Duke Math. J. 44 (1977), 201-213.

[008] [9] W. Rudin, Function Theory in the Unit Ball of $ℂ^{n}$ , Grundlehren Math. Wiss. 241, Springer,1980.

[009] [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. | Zbl 0351.31007

[010] [11] N. Varopoulos, BMO functions and the $\bar{\partial}$ -equation, Pacific J. Math. 71 (1977), 221-273.