On Analytic Functions of Bergman BMO in the Ball

Kwon, E. G.

Canadian mathematical bulletin, Tome 42 (1999) no. 1, pp. 97-103

Résumé

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.

Détail

DOI : 10.4153/CMB-1999-011-3

Mots-clés : 32A37, Bergman distance, BMOA, Hardy space, Bloch function