Simply interpolating and Carleson sequences for Hardy spaces in the polydisc

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 $d\ge 2$, Amar showed that Shapiro and Shields' theorem holds for $H^p({\mathbb{D}}^d)$ when $p\ge 4$. In contrast, we show that if $1\le p\le 2$ there exist simply interpolating sequences which are not universally interpolating.