One-box conditions for Carleson measures for the Dirichlet space

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 $\mu (S(I))=O(\varphi (|I|))$, where $\varphi :(0,2\pi ]\to (0,\infty )$ is an increasing function such that ${\int }_0^{2\pi }(\varphi (x)/x)dx<\infty$. We further show that the integral condition on $\varphi$ is sharp.