Carleson测度与一致有界特征函数

Carleson Measure and the Functions of Uniformly Bounded Characteristic

UBC类与UBC_0类函数分别是BMOA类与VMOA类函数的亚纯推广。我们知道,对单位圆盘上的每一个解析函数,f(z),f(z)∈BMOA当且仅当(1-|z|~2)|f'(z)|~2dxdy是Carleson测度;f(z)∈VMOA当且仅当(1-|z|~2)|f'(z)|~2dxdy是消失Carleson测度。本文我们证明,对单位圆盘上的亚纯函数f(z),f(z)∈UBC_0当且仅当(1-|z|~2)f^(#2)(z)dxdy是消失Carleson测度;若f(z)∈N,则f(z)∈UBC当且仅当(1-|z|~2)f^(#2)(z)dxdy是Carleson测度;其中f~#(z)(?)|f'(z)|/(1+|f(z)|~2)。

BMOA and VMOA have been extended to the meromorphic functions as UBC and UBC_0 respectively.It is well known that for each analytic function f(z) on the unite opendisc, the followings are true: f(z)∈BMOA if and only if (1-|z|~2)|f'(z)|~2dxdy is a Carleson measure; f(z)∈VMOA if and only if (1- |z|~2)|f'(z)|~2 dxdy is a Vanishing Carleson measure.In this paper, we proved the following results, for each meromorphic function f(z) on the unite open disc, f(z)∈UBC_0 if and only if f^(#2)(z) (l-|z|~2)dxdy is a Vanishing Carleson measure; When f(z)∈N, then f(z)∈UBC if and only if f^(#2)(z) (1- |z|~2)dxdy is a Carleson measure; where f~#(z) is defined to be |f'(z)|