Geodesic
On Analytic Functions of Bergman BMO in the Ball
Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 97-103

Voir la notice de l'article provenant de la source Cambridge University Press

Let $B\,=\,{{B}_{n}}$ be the open unit ball of ${{C}^{n}}$ with volume measure $v$ , $U\,=\,{{B}_{1}}$ and $B$ be the Bloch space on $U.\,{{\mathcal{A}}^{2,\alpha }}(B)$ , $1\,\le \,\alpha \,<\infty $ , is defined as the set of holomorphic $f\,:\,B\,\to \,C$ for which $$\int_{B\,}{{{\left| f(z) \right|}^{2}}}{{\left( \frac{1}{\left| z \right|}\,\log \frac{1}{1\,-\left| z \right|} \right)}^{-\alpha }}\,\frac{dv(z)}{1\,-\,\left| z \right|}\,<\,\infty $$ if $0\,<\,\alpha \,<\infty $ and ${{\mathcal{A}}^{2,1}}(B)\,=\,{{H}^{2}}(B)$ , the Hardy space. Our objective of this note is to characterize, in terms of the Bergman distance, those holomorphic $f\,:\,B\,\to \,U$ for which the composition operator ${{C}_{f}}\,:\,B\,\to \,{{\mathcal{A}}^{2,\alpha }}(B)$ defined by ${{C}_{f}}(g)=g\,\text{o}\,f\text{,}\,g\in B$ , is bounded. Our result has a corollary that characterize the set of analytic functions of bounded mean oscillation with respect to the Bergman metric.
DOI : 10.4153/CMB-1999-011-3
Mots-clés : 32A37, Bergman distance, BMOA, Hardy space, Bloch function 
Kwon, E. G. On Analytic Functions of Bergman BMO in the Ball. Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 97-103. doi: 10.4153/CMB-1999-011-3
@article{10_4153_CMB_1999_011_3,
     author = {Kwon, E. G.},
     title = {On {Analytic} {Functions} of {Bergman} {BMO} in the {Ball}},
     journal = {Canadian mathematical bulletin},
     pages = {97--103},
     year = {1999},
     volume = {42},
     number = {1},
     doi = {10.4153/CMB-1999-011-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-011-3/}
}
TY  - JOUR
AU  - Kwon, E. G.
TI  - On Analytic Functions of Bergman BMO in the Ball
JO  - Canadian mathematical bulletin
PY  - 1999
SP  - 97
EP  - 103
VL  - 42
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-011-3/
DO  - 10.4153/CMB-1999-011-3
ID  - 10_4153_CMB_1999_011_3
ER  - 
%0 Journal Article
%A Kwon, E. G.
%T On Analytic Functions of Bergman BMO in the Ball
%J Canadian mathematical bulletin
%D 1999
%P 97-103
%V 42
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-011-3/
%R 10.4153/CMB-1999-011-3
%F 10_4153_CMB_1999_011_3

[CRW] [CRW] Coifman, R., Rochberg, R. and Weiss, G., Factorization theorems for Hardy spaces in several variables. Ann. of Math. 103 (1976), 611–635. Google Scholar

[G] [G] Garnett, J. B., Bounded analytic functions. Academic Press, New York, 1981. Google Scholar

[K1] [K1] Kwon, E. G., Composition of Blochs with bounded analytic functions. Proc. Amer. Math. Soc. 124 (1996), 1473–1480. Google Scholar

[K2] [K2] Kwon, E. G., Mean growth of the hyperbolic Hardy class functions. Math. Japon. 35 (1990), 451–460. Google Scholar

[P] [P] Pavlović, M., Inequalities for the gradient of eigenfunctions of the invariant Laplacian. Indag. Math. (1) 2 (1991), 89–98. Google Scholar

[R] [R] Rudin, W., Function Theory in the unit ball of C . Springer-Verlag, New York, 1980. Google Scholar

[RU] [RU] Ramey, Wade and Ullrich, David, Bounded mean oscillations of Bloch pullbacks. Math. Ann. 291 (1991), 591–606. Google Scholar

[Y1] [Y1] Yamashita, Shinji, Hyperbolic Hardy class H1 . Math. Scand. 45 (1979), 261–266. Google Scholar

[Y2] [Y2] Yamashita, Shinji, Holomorphic functions of hyperbolically bounded mean oscillation. Boll. Un. Mat. Ital. B (6) 5 (1986), 983–1000. Google Scholar

Cité par Sources :