多圆盘上Nehari定理的一个下界
A lower bound in Nehari's theorem on the polydisc
摘要：
根据Ferguson和Lacey（d=2）以及Lacey和Terwilleger（d>2）的定理，已知Nehari定理在多圆盘D^d（d>1）上成立，即，如果H_n是H ^2（D^d）上的有界Hankel形式，其解析符号为H_n，则存在L^ \infty（\T^d）中的函数H_n使得H_n是H_n的Riesz投影. Helson的最后一篇论文中提出的一种方法被用来证明在估计\| \infty \infty \le C_d \|H_d\|中的常数C_d至少随d呈指数增长;由此得出在无限维多圆盘上没有Nehari定理的类似物。
英文摘要：
By theorems of Ferguson and Lacey (d=2) and Lacey and Terwilleger (d>2), Nehari's theorem is known to hold on the polydisc D^d for d>1, i.e., if H_ψis a bounded Hankel form on H^2(D^d) with analytic symbol ψ, then there is a function ϕin L^\infty(\T^d) such that ψis the Riesz projection of ϕ. A method proposed in Helson's last paper is used to show that the constant C_d in the estimate \|ϕ\|_\infty\le C_d \|H_ψ\| grows at least exponentially with d; it follows that there is no analogue of Nehari's theorem on the infinite-dimensional polydisc.
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