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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Estimates of gradient and of Jacobian of harmonic mappings defined in the unit disk


Author: David Kalaj
Journal: Proc. Amer. Math. Soc. 139 (2011), 2463-2472
MSC (2010): Primary 31A05
Published electronically: December 3, 2010
MathSciNet review: 2784812
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Abstract: Let $ F:\mathbb{T}\to \gamma$ be a bounded measurable function of the unit circle $ \mathbb{T}$ onto a rectifiable Jordan curve $ \gamma$ with the length $ \vert\gamma\vert$, and let $ w=P[F]$ be its harmonic extension to the unit disk $ \mathbb{U}$. By using the arc length parametrization of $ \gamma$ we obtain the following results: (i) If $ F$ is a quasi-homeomorphism and $ 1\le p<2$, the $ L^p$-norm of the Hilbert-Schmidt norm of the gradient of $ w$ is bounded as follows: $ \Vert D(w)\Vert _{p}\le \frac{\vert\gamma\vert}{4\sqrt 2}(\frac{16}{\pi(2-p)})^{1/p}$. (ii) If $ F$ is $ p$-Lipschitz continuous and $ \gamma$ is Dini smooth, then the Jacobian of $ w$ is bounded in $ \mathbb{U}$ by a constant $ C(p,\gamma)$. The first result is an extension of a recent result of Verchota and Iwaniec, and Martin and Sbordone, while the second result is an extension of a classical result of Martio where $ \gamma=\mathbb{T}$.


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Additional Information

David Kalaj
Affiliation: Faculty of Natural Sciences and Mathematics, University of Montenegro, Cetinjski put b.b., 81000, Podgorica, Montenegro
Address at time of publication: Faculty of Natural Sciences and Mathematics, University of Montenegro, Džordža Vašingtona b.b., 81000, Podgorica, Montenegro
Email: davidk@t-com.me

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10691-3
PII: S 0002-9939(2010)10691-3
Keywords: Harmonic extension, arc length parametrization, Dirichlet integral
Received by editor(s): December 1, 2009
Received by editor(s) in revised form: December 16, 2009, and June 19, 2010
Published electronically: December 3, 2010
Communicated by: Mario Bonk
Article copyright: © Copyright 2010 American Mathematical Society