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