Extremal functions in invariant subspaces of Bergman spaces
Extremal functions in invariant subspaces of Bergman spaces
复制标题
Bergman 空间不变子空间中的极值函数
DOI：
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发表时间：
1996
期刊：
影响因子：
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通讯作者：
H. Shapiro
中科院分区：
文献类型：
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作者：
P. Duren;D. Khavinson;H. Shapiro
其中Dcr+/da表示面积的归一化元素。Bergman空间理论的最新发展是构造压缩零因子，它是由一个极值问题产生的Blaschke积的弱类似物，该极值问题类似于导致Hardy空间H‘中的Blaschke积的问题。具体地说，如果{‘j是给定的AP零集，且对所有j都有{j#0，且Np是At’中至少在{‘j上以规定的重数或更高的重数消失的所有函数f的子空间，则标准因子G是极值问题的唯一正则解。对于所有的fe np，都是f lip。在零点集的情况下，进一步证明了G具有这样的结构
where dcr +/dA denotes the normalized element of area. A recent development in the theory of Bergman spaces was the construction of contractive zero-divisors, weak analogues of Blaschke products produced by an extremal problem analogous to one that leads to Blaschke products in the Hardy spaces H’. Specifically if {’j is a given AP zero-set with {j # 0 for all j, and NP is the subspace of all functions f in At’ which vanish at least on {’j with the prescribed multiplicity or higher, then the canonical divisor G is the unique normalized solution to the extremal problem of maximizing If(0)l among all f e Np with Ilfllp 1. It is known [6], [3], [4] that the canonical divisor has no extraneous zeros, that Gf II >_ f I1 for all f e AP, and that f/G p < f lip for all f e NP. In the case offinite zero-sets, it was shown further [3] that G has the structure