Simply interpolating and Carleson sequences for Hardy spaces in the polydisc

TL;DR

This paper investigates the differences between simply and universally interpolating sequences in Hardy spaces on the polydisc, revealing that the equivalence in dimension one does not extend to higher dimensions for certain p-values.

Contribution

It demonstrates that for 1 ≤ p ≤ 2 in higher dimensions, simply interpolating sequences are not necessarily universally interpolating, contrasting with the classical dimension one case.

Findings

In dimension one, simply and universally interpolating sequences coincide.

For p between 1 and 2 in higher dimensions, these classes differ.

The classical theorem of Shapiro and Shields does not hold for p in [1,2] when dimension ≥ 2.

Abstract

We study the relation between simply and universally interpolating sequences for the holomorphic Hardy spaces $H^p(\mathbb{D}^d)$ on the polydisc. In dimension $d=1$ a sequence is simply interpolating if and only if it is universally interpolating, due to a classical theorem of Shapiro and Shields. In dimension $d\ge2$, Amar showed that Shapiro and Shields' theorem holds for $H^p(\mathbb{D}^d)$ when $p \geq 4$. In contrast, we show that if $1\leq p \leq 2$ there exist simply interpolating sequences which are not universally interpolating.