One-box conditions for Carleson measures for the Dirichlet space

El-Fallah, Omar; Kellay, Karim; Mashreghi, Javad; Ransford, Thomas

doi:10.1090/S0002-9939-2014-12248-9

One-box conditions for Carleson measures for the Dirichlet space
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by Omar El-Fallah, Karim Kellay, Javad Mashreghi and Thomas Ransford

Proc. Amer. Math. Soc. 143 (2015), 679-684

DOI: https://doi.org/10.1090/S0002-9939-2014-12248-9

Published electronically: September 18, 2014

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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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