Bloch空间与BMOA空间的等价性

EQUIVALENCE BETWEEN BLOCH SPACE AND BMOA SPACE

设Ω是有限复平面C上的双曲型区域,λ_Ω(z)为其上的曲率为-4的Poincare度量并且δ_Ω(z)=dist(z,αΩ)、用B(Ω),QB_k(Ω),BMOA(Ω,m)和BMOH(Ω,m)分别表示Ω上的Bloch空间,拟梯度函数空间,解析的面积BMO空间和实值调和的面积BMO空间.本文证明:在infλ_Ω(m←Ω)(z)δ_Ω(z)>0下,4条等价:(i)f∈B(Ω);(ii)Ref∈BMOH(Ω,m);(iii)f∈BMOA(Ω,m);(iv)Ref∈QB_k(Ω).

Let Ω be a hyperbolic domain in the finite complex plane C, λ_(Ω)(z) be poincare metric with curvature—4 on Ω and δ_(Ω)(z):=dist(z,(α)Ω). Set B(Ω),QB_k(Ω), BMOA(Ω,m) and BMOH(Ω,m) to denote the space of Bloch functions,quasi-gradient functions,analytic area BMO functions and real-valued harmonic area BMO functions on Ω,respctively. In this paper we prove that the followings are equivalent under infλ_(Ω)(z)δ_(Ω)(z)>0, (i)f∈B(Ω); (ii)Ref∈BMOH(Ω,m); (iii)f∈BMOA(Ω,m);(Ⅳ)Ref∈QB_k(Ω).