On the linear extension property for interpolating sequences

TL;DR

This paper proves that sequences interpolating in Hardy and Bergman spaces within the unit ball or polydisc have the bounded linear extension property, using a vectorial approach involving Hilbertian and Besselian bases.

Contribution

It establishes the bounded linear extension property for interpolating sequences in Hardy and Bergman spaces, extending previous results with a vectorial perspective.

Findings

Sequences in Hardy spaces with p ≥ 2 have the bounded linear extension property.

The same property holds for Bergman spaces of the ball using the Subordination Lemma.

A vectorial approach with Hilbertian and Besselian bases is employed.

Abstract

Let $S$ be a sequence of points in $\Omega ,$ where $\Omega$ is the unit ball or the unit polydisc in ${\mathbb{C}}^{n}.$ Denote $H^{p}$($\Omega $) the Hardy space of $\Omega .$ Suppose that $S$ is $H^{p}$ interpolating with $p\geq 2.$ Then $S$ has the bounded linear extension property. The same is true for the Bergman spaces of the ball by use of the "Subordination Lemma". The point of view used here is the vectorial one: Hilbertian and Besselian basis.