Sur Les Suites D'Interpolation Pour Les Espaces De Bergman a Poids Dans la Boule De $\mathbb{C}^n$}

TL;DR

This paper establishes necessary conditions for sequences to be interpolating in weighted Bergman spaces on the unit ball in complex n-space, and addresses the vectorial Gleason problem in these spaces.

Contribution

It provides a new necessary condition for interpolation sequences in weighted Bergman spaces and solves the vectorial Gleason problem within these spaces.

Findings

Derived a necessary condition for interpolation in weighted Bergman spaces.

Solved the vectorial Gleason problem in $B^p_eta (B^n)$.

In the Hardy space case, the condition is also sufficient.

Abstract

Let $A$ be a sequence of points of $\mathbb{B}^n$ the unit ball in $\mathbb{C}^n.$ In terms of interpolating vectorial function (or Amar's function)[1], we give a necessary condition on $A$ to be interpolating for the weighted Bergman space $B^p_\alpha (\mathbb{B}^n). In the particular case of Hardy space $H^p (\mathbb{B}^2)$, this condition is sufficient no optimal. In the main theorem proof, we resolve Gleason's problem (vectorial form) in $B^p_\alpha (\mathbb{B}^n)$