Onto Interpolation for the Dirichlet Space and for $H_1(\mathbb{D})$

TL;DR

This paper characterizes onto interpolating sequences in the Dirichlet space and $H_1(D)$ using condenser capacity, linking interpolation problems in these spaces.

Contribution

It introduces a unified capacity-based criterion for onto interpolation in both the Dirichlet space and $H_1(D)$, extending previous results.

Findings

Capacity condition characterizes onto interpolating sequences.

Interpolation in the Dirichlet space reduces to $H_1(D)$.

Unified framework for interpolation in different function spaces.

Abstract

We give a characterization of onto interpolating sequences with finite associated measure for the Dirichlet space in terms of condenser capacity. In the Sobolev space $H_1(\mathbb{D})$ we define a natural notion of onto interpolation and we prove that the same condenser capacity condition characterizes all onto interpolating sequences. As a result, for sequences with finite associated measure, the problem of interpolation by an analytic function reduces to a problem of interpolation by a function in $H_1(\mathbb{D})$.