论文标题
可插值和二元性在伪内函数的空间中
Interpolation and duality in spaces of pseudocontinuable functions
论文作者
论文摘要
给定单位磁盘上的内部函数$θ$，令$ k^p_θ：= h^p \capθ\ bar z \ bar {h^p} $是hardy space $ h^p $的关联的恒星不变子空间。另外，我们将$ k _ {*θ}放置：= k^2_θ\ cap {\ rm bmo} $。假设$ b = b = b _ {\ Mathcal z} $是一种与零$ \ MATHCAL z = \ {z_j \} $的插值blaschke产品，我们对许多平滑度$ x $，值$ x $，值$ \ \ \ \ \ \ \ \ m artcal w = \ {w_j \ {w_j \ { $ f \ big | _ {\ Mathcal Z} = \ Mathcal W $具有$ k^2_b \ cap x $的解决方案$ f $。转向一般内部函数$θ$的情况，我们进一步建立了$ k^1_θ$和$ k _ {*θ} $之间的非二元关系。也就是说，我们证明后一个空间已正确包含在前者的双重中，除非$θ$是有限的blaschke产品。由此，我们得出了$ k _ {*b} $功能的有趣的非间隔结果，上面是$ b = b = b _ {\ mathcal z} $。
Given an inner function $θ$ on the unit disk, let $K^p_θ:=H^p\capθ\bar z\bar{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*θ}:=K^2_θ\cap{\rm BMO}$. Assuming that $B=B_{\mathcal Z}$ is an interpolating Blaschke product with zeros $\mathcal Z=\{z_j\}$, we characterize, for a number of smoothness classes $X$, the sequences of values $\mathcal W=\{w_j\}$ such that the interpolation problem $f\big|_{\mathcal Z}=\mathcal W$ has a solution $f$ in $K^2_B\cap X$. Turning to the case of a general inner function $θ$, we further establish a non-duality relation between $K^1_θ$ and $K_{*θ}$. Namely, we prove that the latter space is properly contained in the dual of the former, unless $θ$ is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in $K_{*B}$, with $B=B_{\mathcal Z}$ as above.
