We show that if $\gamma$ is a curve in the unit disk, then arclength on $\gamma$ is a Carleson measure iff the image of $\gamma$ has finite length under every conformal map of the disk onto a bounded domain with a rectifiable boundary.
In this note we characterize curves in $\mathbb{D}$ for which arclength is a Carleson measure, in terms of conformal maps onto rectifiable domains, answering a question asked by Percy Deift (personal communication) arising from his work on Riemann-Hilbert problems. The question seems natural and the proof follows from standard techniques, but I have not been able to locate this result in the literature.
Recall that a positive measure $\mu$ on the open unit disk, $\mathbb{D}$, is called a Carleson measure if
The left hand side is called the Carleson norm of the measure.
Although Deift’s question concerned curves, we never used this, and we have actually proven that a positive measure $\mu$ on the disk is Carleson iff $\int |f'|d \mu < \infty$ for any conformal map $f$ onto a rectifiable domain.
Acknowledgments
I thank the anonymous referee for several helpful comments and suggestions that clarified the argument and improved the exposition of this note. Also thanks to Percy Deift for raising the problem originally and encouraging me to record its solution.
The top picture shows the domain $\Omega _{a,\epsilon }$ which is a small disk attached to the unit disk. A properly placed Carleson region is expanded by this map to a size comparable to the added “bubble” and $|f'|$ is comparable to the ratio of the diameters of the region and its image. By composing maps of this form, we build a sequence of domains that look like the lower picture, except that in the proof the sizes of the “bubbles” shrink much more dramatically.
References
Reference [1]
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR628971, Show rawAMSref\bib{MR628971}{book}{
author={Garnett, John B.},
title={Bounded analytic functions},
series={Pure and Applied Mathematics},
volume={96},
publisher={Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London},
date={1981},
pages={xvi+467},
isbn={0-12-276150-2},
review={\MR {628971}},
}
Reference [2]
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2008. Reprint of the 2005 original. MR2450237, Show rawAMSref\bib{MR2450237}{book}{
author={Garnett, John B.},
author={Marshall, Donald E.},
title={Harmonic measure},
series={New Mathematical Monographs},
volume={2},
note={Reprint of the 2005 original},
publisher={Cambridge University Press, Cambridge},
date={2008},
pages={xvi+571},
isbn={978-0-521-72060-1},
review={\MR {2450237}},
}
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