Pointwise multipliers and corona type decomposition in $BMOA$

Ortega, J. M.; Fàbrega, Joan

doi:10.5802/aif.1509

Pointwise multipliers and corona type decomposition in

B M O A

Ortega, J. M. ; Fàbrega, Joan

Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 111-137.

Résumé
Abstract

Dans cet article nous obtenons plusieurs caractérisations des multiplicateurs ponctuels de l’espace $B M O A$ dans la boule unité $B$ de $ℂ^{n}$ . De plus, si $g_{1}, ..., g_{m}$ sont des fonctions holomorphes dans $B$ nous démontrons que l’application $M_{g} (f) (z) = \sum_{j = 1}^{m} g_{j} (z) f_{j} (z)$ envoie $B M O A \times ... \times B M O A$ sur $B M O A$ si et seulement si les fonctions $g_{j}$ sont multiplicateurs de l’espace $B M O A$ et satisfont $\sum_{j = 1}^{m} | g_{j} (z) | \geq δ > 0 .$

In this paper we obtain several characterizations of the pointwise multipliers of the space $B M O A$ in the unit ball $B$ of $ℂ^{n}$ . Moreover, if $g_{1}, ..., g_{m}$ are holomorphic functions on $B$ , we prove that $M_{g} (f) (z) = \sum_{j = 1}^{m} g_{j} (z) f_{j} (z)$ maps $B M O A \times ... \times B M O A$ onto $B M O A$ if and only if the functions $g_{j}$ are multipliers of the space $B M O A$ and satisfy $\sum_{j = 1}^{m} | g_{j} (z) | \geq δ > 0 .$

MR Zbl

DOI : 10.5802/aif.1509

@article{AIF_1996__46_1_111_0,
     author = {Ortega, J. M. and F\`abrega, Joan},
     title = {Pointwise multipliers and corona type decomposition in $BMOA$},
     journal = {Annales de l'Institut Fourier},
     pages = {111--137},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     doi = {10.5802/aif.1509},
     zbl = {0840.32001},
     mrnumber = {97k:32009},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1509/}
}

TY  - JOUR
AU  - Ortega, J. M.
AU  - Fàbrega, Joan
TI  - Pointwise multipliers and corona type decomposition in $BMOA$
JO  - Annales de l'Institut Fourier
PY  - 1996
SP  - 111
EP  - 137
VL  - 46
IS  - 1
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1509/
DO  - 10.5802/aif.1509
LA  - en
ID  - AIF_1996__46_1_111_0
ER  -

%0 Journal Article
%A Ortega, J. M.
%A Fàbrega, Joan
%T Pointwise multipliers and corona type decomposition in $BMOA$
%J Annales de l'Institut Fourier
%D 1996
%P 111-137
%V 46
%N 1
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1509/
%R 10.5802/aif.1509
%G en
%F AIF_1996__46_1_111_0

Ortega, J. M.; Fàbrega, Joan. Pointwise multipliers and corona type decomposition in $BMOA$. Annales de l'Institut Fourier, Tome 46 (1996) no. 1, pp. 111-137. doi : 10.5802/aif.1509. https://aif.centre-mersenne.org/articles/10.5802/aif.1509/

Bibliographie
Cité par

[A] E. Amar, On the corona theorem., J. of Geom. Anal., 1 (1991), 291-305. | MR | Zbl

[AB] E. Amar, A. Bonami, Mesures de Carleson d'ordre α et estimations de solutions du ∂, BSM France, 107 (1979), 23-48. | EuDML | Numdam | MR | Zbl

[An] M. Andersson, On the Hp-corona problem, Bull. Sci. Math., 118 (1994), 287-306. | MR | Zbl

[AnCa1] M. Andersson, H. Carlsson, Wolff-type estimates and the Hp-corona problem in strictly pseudoconvex domains, Ark. Mat., 32 (1994), 255-276. | MR | Zbl

[AnCa2] M. Andersson, H. Carlsson, Hp-estimates of holomorphic division formulas, Preprint. | Zbl

[AxSh] Sh. Axler, J. Shapiro, Putnam's theorem, Alexander's spectral area estimate and VMO, Math. Ann., 271 (1985), 161-183. | EuDML | MR | Zbl

[B] B. Berndtsson, A formula for division and interpolation, Math. Ann., 263 (1983), 399-418. | EuDML | MR | Zbl

[BAn] B. Berndtsson, M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble), 32-3 (1983), 91-110. | EuDML | Numdam | MR | Zbl

[BrBu] J. Bruna, J.M. Burgués, Holomorphic Approximation in Cm-Norms on Totally Real Compact Sets in ℂn, Math. Ann., 269 (1984), 103-117. | EuDML | MR | Zbl

[C] L. Carleson, Interpolation by bounded analytic functions and the corona theorem, Ann. of Math., 76 (1962), 547-559. | MR | Zbl

[Ch] P. Charpentier, Solutions minimales de l'équation ∂u = f dans la boule et le polydisque, Ann. Inst. Fourier (Grenoble), 30-4 (1980), 121-153. | Numdam | MR | Zbl

[Chee] P.S. Chee, VMOA and vanishing Carleson measures, Complex Variables Theory and Appl., 25 (1994), 311-322. | MR | Zbl

[ChoaChoe] J.S. Choa, B.R. Choe, A Littlewwod-Paley type identity and a characterization of BMOA, Complex Variables Theory and Appl., 17 (1991), 15-23. | MR | Zbl

[CoRoW] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems in Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-645. | MR | Zbl

[CoW] R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. | MR | Zbl

[Cu] A. Cumenge, Extension holomorphe dans des classes de Hardy et estimations de type mesures de Carleson pour ∂u = f, Ann. Inst. Fourier, 33-3 (1983), 59-97. | Numdam | MR | Zbl

[G] J.B. Garnett, Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981. | MR | Zbl

[H] L. Hormander, An introduction to complex analysis in several variables, Van Nostram, Princeton, 1966. | MR | Zbl

[J] M. Jevtic, A note on the Carleson measure characterization of BMOA functions on the unit ball, Complex Variables Theory and Appl., 25 (1992), 189-194. | MR | Zbl

[Li] S.Y. Li, Corona problem of several complex variables, Contemporary Math., 137 (1992), 307-328. | MR | Zbl

[Lin] K.C. Lin, Hp solutions for the corona problem on the polidisc of ℂn, Bull. A.M.S., 110 (1986), 69-84. | MR | Zbl

[MS] V.G. Maz'Da, T.O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Pitmann, 1985. | Zbl

[N] A. Nicolau, The corona property for bounded analytic functions in some Besov spaces, Proc. Amer. Math. Soc., 110 (1990), 135-140. | MR | Zbl

[OF] J.M. Ortega, J. Fàbrega, Corona type decomposition in some Besov spaces, to appear in Math. Scand. | Zbl

[R] W. Rudin, Function theory in the unit ball of ℂn, Springer-Verlag, New York, 1980. | MR | Zbl

[Sa] D. Sarason, Function theory in the unit circle, Virginia Poly. Inst. and St. Univ., Backburg, VA, 1979. | Zbl

[Sh] J. Shapiro, Cluster sets, essential range and distance estimates in BMO, Mich. Math. J., 34 (1987), 323-335. | MR | Zbl

[S] D. Stegenga, Bounded toeplitz operators on H1 and application of the duality between H1 and the functions of Bounded Mean Oscilation, Amer. J. Math., 98 (1976), 573-589. | MR | Zbl

[T] V.A. Tolokonnikov, The corona theorem in algebras of bounded analytic functions, Amer. Math. Soc. Trans., 149 (1991), 61-93. | Zbl

[V1] N. Varopoulos, BMO functions and the ∂-equation, Pacific J. Math., 71 (1977), 221-273. | MR | Zbl

[V2] N. Varopoulos, Zeros of Hp functions in several complex variables, Pacific J. Math., 88 (1980), 189-246. | MR | Zbl

[Z] K. Zhu, Multipliers of BMO in the Bergman metric with applications to Toeplitz operators, J. Functional Anal., 87 (1989), 31-50. | MR | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT