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The Bloch functions in the unit ball of \({\mathbb{C}}^ n\). (Chinese. English summary) Zbl 0628.32007

This paper studies properties of Bloch functions in the unit ball B of \({\mathbb{C}}^ n\) concerning the following aspects: the growth order, the coefficients of the Taylor expansions, the relations between the Bloch functions and \(H^ p\) functions, and the Zygmund condition for the Bloch functions. As an application, it is proved that the two classes of BMOA functions in B, defined by the Euclidean metric and the Bergman metric respectively, are exactly the same. This result is interesting compared with the case on \(\partial B\). Finally, a relationship is given between the Bloch functions and BMOA(\(\partial B)\), the dual space of \(H^ 1(B)\), which enables us to construct a BMOA(\(\partial B)\) function such that its radial limits do not exist almost everywhere on a submanifold of dimension \((2n-3)\) of \(\partial B\).

MSC:

32A10 Holomorphic functions of several complex variables
32A30 Other generalizations of function theory of one complex variable
30D50 Blaschke products, etc. (MSC2000)
32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
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