One-box conditions for Carleson measures for the Dirichlet space
Résumé
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 $\phi: (0, 2 ] \to (0,1)$ 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.