Pointwise multipliers and corona type decomposition in BMOA
Annales de l'Institut Fourier, Tome 46 (1996) no. 1, p. 111-137
Dans cet article nous obtenons plusieurs caractérisations des multiplicateurs ponctuels de l’espace BMOA dans la boule unité B de n . De plus, si g 1 ,...,g m sont des fonctions holomorphes dans B nous démontrons que l’application M g (f)(z)= j=1 m g j (z)f j (z) envoie BMOA×...×BMOA sur BMOA si et seulement si les fonctions g j sont multiplicateurs de l’espace BMOA et satisfont j=1 m |g j (z)|δ>0.
In this paper we obtain several characterizations of the pointwise multipliers of the space BMOA in the unit ball B of n . Moreover, if g 1 ,...,g m are holomorphic functions on B, we prove that M g (f)(z)= j=1 m g j (z)f j (z) maps BMOA×...×BMOA onto BMOA if and only if the functions g j are multipliers of the space BMOA and satisfy j=1 m |g j (z)|δ>0.
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {46},
     number = {1},
     year = {1996},
     pages = {111-137},
     doi = {10.5802/aif.1509},
     zbl = {0840.32001},
     mrnumber = {97k:32009},
     language = {en},
     url = {http://www.numdam.org/item/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. http://www.numdam.org/item/AIF_1996__46_1_111_0/

[A] E. Amar, On the corona theorem., J. of Geom. Anal., 1 (1991), 291-305. | MR 92h:32006 | Zbl 0794.32007

[AB] E. Amar, A. Bonami, Mesures de Carleson d'ordre α et estimations de solutions du ∂, BSM France, 107 (1979), 23-48. | Numdam | MR 80h:32032 | Zbl 0409.46035

[An] M. Andersson, On the Hp-corona problem, Bull. Sci. Math., 118 (1994), 287-306. | MR 95g:32006 | Zbl 0810.32002

[AnCa1] M. Andersson, H. Carlsson, Wolff-type estimates and the Hp-corona problem in strictly pseudoconvex domains, Ark. Mat., 32 (1994), 255-276. | MR 96j:32022 | Zbl 0827.32017

[AnCa2] M. Andersson, H. Carlsson, Hp-estimates of holomorphic division formulas, Preprint. | Zbl 0857.32006

[AxSh] Sh. Axler, J. Shapiro, Putnam's theorem, Alexander's spectral area estimate and VMO, Math. Ann., 271 (1985), 161-183. | MR 87b:30053 | Zbl 0541.30021

[B] B. Berndtsson, A formula for division and interpolation, Math. Ann., 263 (1983), 399-418. | MR 85b:32005 | Zbl 0499.32013

[BAn] B. Berndtsson, M. Andersson, Henkin-Ramirez formulas with weight factors, Ann. Inst. Fourier (Grenoble), 32-3 (1983), 91-110. | Numdam | MR 84j:32003 | Zbl 0466.32001

[BrBu] J. Bruna, J.M. Burgués, Holomorphic Approximation in Cm-Norms on Totally Real Compact Sets in ℂn, Math. Ann., 269 (1984), 103-117. | MR 86c:32014 | Zbl 0527.32010

[C] L. Carleson, Interpolation by bounded analytic functions and the corona theorem, Ann. of Math., 76 (1962), 547-559. | MR 25 #5186 | Zbl 0112.29702

[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 82j:32009 | Zbl 0425.32009

[Chee] P.S. Chee, VMOA and vanishing Carleson measures, Complex Variables Theory and Appl., 25 (1994), 311-322. | MR 95m:32008 | Zbl 0819.32005

[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 92j:32010 | Zbl 0699.32005

[CoRoW] R. Coifman, R. Rochberg, G. Weiss, Factorization theorems in Hardy spaces in several variables, Ann. of Math., 103 (1976), 611-645. | MR 54 #843 | Zbl 0326.32011

[CoW] R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83 (1977), 569-645. | MR 56 #6264 | Zbl 0358.30023

[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 85j:32005 | Zbl 0487.32011

[G] J.B. Garnett, Bounded Analytic Functions, Academic Press, Orlando, Fla., 1981. | MR 83g:30037 | Zbl 0469.30024

[H] L. Hormander, An introduction to complex analysis in several variables, Van Nostram, Princeton, 1966. | MR 34 #2933 | Zbl 0138.06203

[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 93a:32007 | Zbl 0758.32003

[Li] S.Y. Li, Corona problem of several complex variables, Contemporary Math., 137 (1992), 307-328. | MR 93m:32011 | Zbl 0763.32008

[Lin] K.C. Lin, Hp solutions for the corona problem on the polidisc of ℂn, Bull. A.M.S., 110 (1986), 69-84. | MR 88a:32009 | Zbl 0601.32004

[MS] V.G. Maz'Da, T.O. Shaposhnikova, Theory of multipliers in spaces of differentiable functions, Pitmann, 1985. | Zbl 0645.46031

[N] A. Nicolau, The corona property for bounded analytic functions in some Besov spaces, Proc. Amer. Math. Soc., 110 (1990), 135-140. | MR 90m:46090 | Zbl 0708.46050

[OF] J.M. Ortega, J. Fàbrega, Corona type decomposition in some Besov spaces, to appear in Math. Scand. | Zbl 0853.32002

[R] W. Rudin, Function theory in the unit ball of ℂn, Springer-Verlag, New York, 1980. | MR 82i:32002 | Zbl 0495.32001

[Sa] D. Sarason, Function theory in the unit circle, Virginia Poly. Inst. and St. Univ., Backburg, VA, 1979. | Zbl 0398.30027

[Sh] J. Shapiro, Cluster sets, essential range and distance estimates in BMO, Mich. Math. J., 34 (1987), 323-335. | MR 89b:30029 | Zbl 0634.30033

[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 54 #8340 | Zbl 0335.47018

[T] V.A. Tolokonnikov, The corona theorem in algebras of bounded analytic functions, Amer. Math. Soc. Trans., 149 (1991), 61-93. | Zbl 0765.46041

[V1] N. Varopoulos, BMO functions and the ∂-equation, Pacific J. Math., 71 (1977), 221-273. | MR 58 #22639a | Zbl 0371.35035

[V2] N. Varopoulos, Zeros of Hp functions in several complex variables, Pacific J. Math., 88 (1980), 189-246. | MR 82d:32004 | Zbl 0454.32006

[Z] K. Zhu, Multipliers of BMO in the Bergman metric with applications to Toeplitz operators, J. Functional Anal., 87 (1989), 31-50. | MR 90m:42024 | Zbl 0705.47025