On separated Carleson sequences in the unit disc. (English) Zbl 1304.30068

The author obtains a functional characterization of the \(H^\infty\)-interpolating sequences in the unit disk \(\mathbb{D}\) of \(\mathbb{C}\).
Let \(S\) be a sequence of points in \(\mathbb{D}\) and let \((A,B)\) be a restricted good partition or a Hoffman partition of \(S\). For \(\kappa\geq 1\), the sequence \(S\) is \(\kappa\)-ultra-separated if it is separated and there exist constants \(0<\tau<\eta<1\), \(\tau<\eta^\kappa\), and a function \(f\in H^\infty(\mathbb{D})\), such that \(\|f\|_{\infty}\leq 1\), \(|f|\leq \tau\) on \(A\) and \(|f|\geq \eta\) on \(B\).
The main result of the article under review states that there is \(\kappa>1\) such that \(S\) is \(H^\infty\)-interpolating if and only if it is \(\kappa\)-ultra-separated.
This generalizes a result of A. Hartmann [Proc. Am. Math. Soc. 140, No. 7, 2411–2416 (2012; Zbl 1275.30001)], where \(|f|=0\) on \(A\) and \(|f|= \eta\) on \(B\).


30H10 Hardy spaces
30H80 Corona theorems


Zbl 1275.30001
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