# One-box conditions for Carleson measures for the Dirichlet space

## TL;DR

This paper provides a simplified proof that finite measures satisfying a specific one-box Carleson condition are valid Carleson measures for the Dirichlet space, and demonstrates the sharpness of the integral condition on the function involved.

## Contribution

It introduces a straightforward proof of the Carleson measure characterization for the Dirichlet space using a one-box condition and establishes the optimality of the integral condition on the function.

## Key findings

- The one-box Carleson condition characterizes Carleson measures for the Dirichlet space.
- The integral condition on the function  is shown to be sharp.
- The proof simplifies existing arguments for this characterization.

## Abstract

We give a simple proof of the fact that a finite measure $\mu$ on the unit disk is a Carleson measure for the Dirichlet space if it satisfies the Carleson one-box condition $\mu(S(I))=O(\phi(|I|))$, where $\phi:(0,2\pi]\to(0,\infty)$ is an increasing function such that $\int_0^{2\pi}(\phi(x)/x)\,dx<\infty$. We further show that the integral condition on $\phi$ is sharp.