One-box conditions for Carleson measures for the Dirichlet space

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