Sharp distortion growth for bilipschitz extension of planar maps

Kovalev, Leonid

doi:10.1090/S1088-4173-2012-00243-3

Sharp distortion growth for bilipschitz extension of planar maps
HTML articles powered by AMS MathViewer

by Leonid V. Kovalev

Conform. Geom. Dyn. 16 (2012), 124-131

DOI: https://doi.org/10.1090/S1088-4173-2012-00243-3

Published electronically: April 18, 2012

PDF | Request permission

Abstract:

This note addresses the quantitative aspect of the bilipschitz extension problem. The main result states that any bilipschitz embedding of $\mathbb R$ into $\mathbb R^2$ can be extended to a bilipschitz self-map of $\mathbb R^2$ with a linear bound on the distortion.

References

Lars V. Ahlfors, Lectures on quasiconformal mappings, 2nd ed., University Lecture Series, vol. 38, American Mathematical Society, Providence, RI, 2006. With supplemental chapters by C. J. Earle, I. Kra, M. Shishikura and J. H. Hubbard. MR 2241787, DOI 10.1090/ulect/038
P. Alestalo, D. A. Trotsenko, and Yu. Vyaĭsyalya, The linear extension property of bi-Lipschitz mappings, Sibirsk. Mat. Zh. 44 (2003), no. 6, 1226–1238 (Russian, with Russian summary); English transl., Siberian Math. J. 44 (2003), no. 6, 959–968. MR 2034930, DOI 10.1023/B:SIMJ.0000007471.47551.5d
P. Alestalo and D. A. Trotsenko, Plane sets allowing bilipschitz extensions, Math. Scand. 105 (2009), no. 1, 134–146. MR 2549802, DOI 10.7146/math.scand.a-15110
J. Azzam and R. Schul, Hard Sard: quantitative implicit function and extension theorems for Lipschitz maps, Geom. Funct. Anal., arXiv:1105.4198.
A. Beurling and L. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142. MR 86869, DOI 10.1007/BF02392360
A. Brudnyi and Yu. Brudnyi, Methods of geometric analysis in extension and trace problems, Birkhäuser, 2011.
S. Daneri and A. Pratelli, A planar bi-Lipschitz extension theorem, arXiv:1110.6124.
G. David and S. Semmes, Singular integrals and rectifiable sets in $\textbf {R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517
Adrien Douady and Clifford J. Earle, Conformally natural extension of homeomorphisms of the circle, Acta Math. 157 (1986), no. 1-2, 23–48. MR 857678, DOI 10.1007/BF02392590
John B. Garnett and Donald E. Marshall, Harmonic measure, New Mathematical Monographs, vol. 2, Cambridge University Press, Cambridge, 2005. MR 2150803, DOI 10.1017/CBO9780511546617
David S. Jerison and Carlos E. Kenig, Hardy spaces, $A_{\infty }$, and singular integrals on chord-arc domains, Math. Scand. 50 (1982), no. 2, 221–247. MR 672926, DOI 10.7146/math.scand.a-11956
D. Kalaj, Radial extension of a bi-Lipschitz parametrization of a starlike Jordan curve, arXiv:1011.5204.
M. D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen, Fund. Math. 22 (1934), 77–108.
T. G. Latfullin, On geometric conditions for images of lines and circles under planar quasiisometries, in “Proceedings of the XVIII All-Union scientific student conference”, Novosibirsk, 1980, 18–22.
T. G. Latfullin, Continuation of quasi-isometric mappings, Sibirsk. Mat. Zh. 24 (1983), no. 4, 212–216 (Russian). MR 713600
A. Lukyanenko, Bi-Lipschitz extension from boundaries of certain hyperbolic spaces, arXiv:1112.2684.
Paul MacManus, Bi-Lipschitz extensions in the plane, J. Anal. Math. 66 (1995), 85–115. MR 1370347, DOI 10.1007/BF02788819
Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706, DOI 10.1007/978-3-662-02770-7
Thomas Ransford, Potential theory in the complex plane, London Mathematical Society Student Texts, vol. 28, Cambridge University Press, Cambridge, 1995. MR 1334766, DOI 10.1017/CBO9780511623776
Stephen Semmes, Chord-arc surfaces with small constant. II. Good parameterizations, Adv. Math. 88 (1991), no. 2, 170–199. MR 1120612, DOI 10.1016/0001-8708(91)90007-T
D. A. Trotsenko, Extendability of classes of maps and new properties of upper sets, Complex Anal. Oper. Theory 5 (2011), no. 3, 967–984. MR 2836337, DOI 10.1007/s11785-010-0096-z
Pekka Tukia, The planar Schönflies theorem for Lipschitz maps, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 49–72. MR 595177, DOI 10.5186/aasfm.1980.0529
Pekka Tukia, Extension of quasisymmetric and Lipschitz embeddings of the real line into the plane, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 1, 89–94. MR 639966, DOI 10.5186/aasfm.1981.0624
P. Tukia and J. Väisälä, Lipschitz and quasiconformal approximation and extension, Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982). MR 658932, DOI 10.5186/aasfm.1981.0626
P. Tukia and J. Väisälä, Extension of embeddings close to isometries or similarities, Ann. Acad. Sci. Fenn. Ser. A I Math. 9 (1984), 153–175. MR 752401, DOI 10.5186/aasfm.1984.0914
Pekka Tukia and Jussi Väisälä, Bi-Lipschitz extensions of maps having quasiconformal extensions, Math. Ann. 269 (1984), no. 4, 561–572. MR 766014, DOI 10.1007/BF01450765
Jussi Väisälä, Bi-Lipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fenn. Ser. A I Math. 11 (1986), no. 2, 239–274. MR 853960, DOI 10.5186/aasfm.1986.1117