## 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 PDF
- Proc. Amer. Math. Soc.
**143**(2015), 679-684 Request permission

## 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.## References

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## Additional Information

**Omar El-Fallah**- Affiliation: Laboratoire Analyse et Applications (CNRST URAC03), Université Mohamed V, B. P. 1014 Rabat, Morocco
- MR Author ID: 338521
- Email: elfallah@fsr.ac.ma
**Karim Kellay**- Affiliation: IMB, Université Bordeaux 1, 351 cours de la Libération, F-33405 Talence cedex, France
- Email: karim.kellay@math.u-bordeaux1.fr
**Javad Mashreghi**- Affiliation: Département de mathématiques et de statistique, Université Laval, Québec, Canada G1V 0A6
- MR Author ID: 679575
- Email: javad.mashreghi@mat.ulaval.ca
**Thomas Ransford**- Affiliation: Département de mathématiques et de statistique, Université Laval, Québec, Canada G1V 0A6
- MR Author ID: 204108
- Email: ransford@mat.ulaval.ca
- Received by editor(s): February 15, 2013
- Received by editor(s) in revised form: April 30, 2013
- Published electronically: September 18, 2014
- Additional Notes: The first author was supported by Académie Hassan II des sciences et techniques

The second author was supported by PICS-CNRS

The third author was supported by NSERC

The fourth author was supported by NSERC and the Canada research chairs program - Communicated by: Pamela B. Gorkin
- © Copyright 2014 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**143**(2015), 679-684 - MSC (2010): Primary 31C25; Secondary 28C99
- DOI: https://doi.org/10.1090/S0002-9939-2014-12248-9
- MathSciNet review: 3283654