# Carleson measures on simply connected domains

## TL;DR

This paper investigates Carleson measures on simply connected domains, emphasizing the influence of domain geometry on Hardy and weighted Bergman spaces, and revisits classical results with alternative proofs.

## Contribution

It extends the study of Carleson measures to weighted Bergman spaces on general simply connected domains, providing new insights and alternative proofs.

## Key findings

- Geometry of the domain significantly affects Carleson measures.
- Classical results are revisited with alternative proofs.
- Analysis includes weighted Bergman spaces on simply connected domains.

## Abstract

We study the Carleson measures associated to the Hardy and weighted Bergman spaces defined on general simply connected domains. This program was initiated by Zinsmeister in his paper \textit{Les domaines de Carleson }(1989), where he shows that the geometry of the domain plays a fundamental role. We will review the classical results presenting, in some cases, alternative proofs and will examine the situation for the weighted Bergman spaces.