Three Remarks on Carleson Measures for Dirichlet Space

TL;DR

This paper establishes that all doubling measures on the unit disk are Carleson measures for the Dirichlet space, introducing new characterizations and applying dyadic methods to this classical problem.

Contribution

It provides a comprehensive proof that all doubling measures are Carleson measures for the Dirichlet space, with new characterizations specific to this space and a novel application of dyadic techniques.

Findings

All doubling measures on the unit disk are Carleson measures for the Dirichlet space.

New equivalent conditions for Carleson measures are established for the Dirichlet space.

Dyadic methods are effectively applied to analyze Carleson measures in this setting.

Abstract

In this paper, we prove that all doubling measures on the unit disk $\mathbb{D}$ are Carleson measures for the standard Dirichlet space $\mathcal{D}$. The proof has three ingredients. The first one is a characterization of Carleson measures which holds true for general reproducing kernel Hilbert spaces. The second one is another new equivalent condition for Carleson measures, which holds true only for the standard Dirichlet space. The third one is an application of the dyadic method to our settings.