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arXiv · 1902.05932

One-box conditions for Carleson measures for the Dirichlet space

Omar El-Fallah·Karim Kellay·Javad Mashreghi·Thomas Ransford
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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.

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Omar El-Fallah, Karim Kellay, Javad Mashreghi, Thomas Ransford. 2019-02-15. One-box conditions for Carleson measures for the Dirichlet space. https://arxiv.org/abs/1902.05932

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