Estimates of gradient and of Jacobian of harmonic mappings defined in the unit disk
HTML articles powered by AMS MathViewer
- by David Kalaj PDF
- Proc. Amer. Math. Soc. 139 (2011), 2463-2472 Request permission
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 $|\gamma |$, 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: $\|D(w)\|_{p}\le \frac {|\gamma |}{4\sqrt 2}(\frac {16}{\pi (2-p)})^{1/p}$. (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 $\gamma =\mathbb T$.References
- K. Astala, T. Iwaniec, G. J. Martin, and J. Onninen, Extremal mappings of finite distortion, Proc. London Math. Soc. (3) 91 (2005), no. 3, 655–702. MR 2180459, DOI 10.1112/S0024611505015376
- Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic function theory, Graduate Texts in Mathematics, vol. 137, Springer-Verlag, New York, 1992. MR 1184139, DOI 10.1007/b97238
- Daoud Bshouty, Walter Hengartner, and M. Naghibi-Beidokhti, $p$-valent harmonic mappings with finite Blaschke dilatations, Ann. Univ. Mariae Curie-Skłodowska Sect. A 53 (1999), 9–26. XII-th Conference on Analytic Functions (Lublin, 1998). MR 1775530
- Peter Duren, Harmonic mappings in the plane, Cambridge Tracts in Mathematics, vol. 156, Cambridge University Press, Cambridge, 2004. MR 2048384, DOI 10.1017/CBO9780511546600
- David Kalaj, Lipschitz spaces and harmonic mappings, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 2, 475–485. MR 2553807
- D. Kalaj, M. Pavlović, On quasiconformal self-mappings of the unit disk satisfying Poisson’s equation, Trans. Amer. Math. Soc, in press.
- Tadeusz Iwaniec, Gaven Martin, and Carlo Sbordone, $L^p$-integrability & weak type $L^2$-estimates for the gradient of harmonic mappings of $\Bbb D$, Discrete Contin. Dyn. Syst. Ser. B 11 (2009), no. 1, 145–152. MR 2461814, DOI 10.3934/dcdsb.2009.11.145
- O. Martio, On harmonic quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 425 (1968), 10. MR 0236382
- Kurt Strebel, Quadratic differentials, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 5, Springer-Verlag, Berlin, 1984. MR 743423, DOI 10.1007/978-3-662-02414-0
- Gregory C. Verchota, Harmonic homeomorphisms of the closed disc to itself need be in $W^{1,p},\ p<2$, but not $W^{1,2}$, Proc. Amer. Math. Soc. 135 (2007), no. 3, 891–894. MR 2262887, DOI 10.1090/S0002-9939-06-08506-6
- A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776
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
- MR Author ID: 689421
- Email: davidk@t-com.me
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 139 (2011), 2463-2472
- MSC (2010): Primary 31A05
- DOI: https://doi.org/10.1090/S0002-9939-2010-10691-3
- MathSciNet review: 2784812