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One-box conditions for Carleson measures for the Dirichlet space


Authors: Omar El-Fallah, Karim Kellay, Javad Mashreghi and Thomas Ransford
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
Published electronically: September 18, 2014
MathSciNet review: 3283654
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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 (\vert I\vert))$, 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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Additional Information

Omar El-Fallah
Affiliation: Laboratoire Analyse et Applications (CNRST URAC03), Université Mohamed V, B. P. 1014 Rabat, Morocco
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
Email: javad.mashreghi@mat.ulaval.ca

Thomas Ransford
Affiliation: Département de mathématiques et de statistique, Université Laval, Québec, Canada G1V 0A6
Email: ransford@mat.ulaval.ca

DOI: https://doi.org/10.1090/S0002-9939-2014-12248-9
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
Article copyright: © Copyright 2014 American Mathematical Society

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