C^n中单位球上函数空间及其对偶

The Function Spaces on Unit Ball in C n and Its Duality

设和ψ是Ｃｎ中单位球Ｂ上分别满足当｜ｚ｜→１时（ｚ）→０和∫Ｂψｄｖ＜∞的正连续函数。令Ａ０（）和Ａ∞（）分别表示由Ｂ上满足条件当｜ｚ｜→１时，｜ｆ（ｚ）｜（ｚ）→０和｜ｆ（ｚ）｜（ｚ）＜Ｃ（Ｃ为正常数）的解析函数全体按范数‖ｆ‖＝ｓｕｐ｜ｆ（ｚ）（ｚ）｜所成的Ｂａｎａｃｈ空间，Ａ１（ψ）表示Ｂ上解析函数全体按范数‖ｆ‖＝∫Ｂ｜ｆ｜ψｄｖ＜∞所成的空间。该文指出如果权函数，ψ为一正规对，那么Ａ０（）的对偶空间拓朴同构于Ａ１（ψ），以及Ａ∞（）拓朴同构于Ａ１（ψ）的对偶空间。

Let  and ψ be positive continnous functio ns on B , the unit ball in C n , with (z)→0 as |z|→1 and ∫ Bψdv<∞ . Denote A 0() and A ∞() the Banach spaces of fu nctions f holomorphic in B with |f(z)|(z)→0 as |z|→1 and |f(z)|(z)<C for some constant C>0 respectively, using the norm ‖f‖= sup |f(z)(z)|. Let A 1(ψ) denote the space of holomorphic functions on B with ‖f‖=∫ B|f|ψdv<∞ . We show that if  and ψ is a pair of weight functions which we define as normal pair, the dual of A 0() is topologically isormorphic to A 1(ψ) , and A ∞() is topologically isomorphic to the dual of A 1(ψ) .