On smoothness of symmetric mappings II
Author:
A. Cantón
Journal:
Proc. Amer. Math. Soc. 133 (2005), 103113
MSC (2000):
Primary 30C62; Secondary 30E25
Published electronically:
June 2, 2004
MathSciNet review:
2085159
Fulltext PDF Free Access
Abstract 
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Abstract: If the dilatation of a quasiconformal selfmap of the upper halfplane vanishes near the real line as a power of the height, the induced quasisymmetric mapping is Lipschitz with the same exponent. In this note, it is shown that the converse does not hold for any positive exponent. In addition, a sufficient condition is found to have locally a quasiconformal extension with the desired growth in the dilatation.
 1.
J.M. Anderson, A. Cantón and J.L. Fernández, On smoothness of symmetric mappings, Complex Var. Theory Appl., 37 (1998), 161169.
 2.
J.M. Anderson and A. Hinkkanen, Quasiconformal selfmappings with smooth boundary values, Bull. London Math. Soc., 26 (1994), 549556.MR 96a:30018
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K.E. Atkinson, An Introduction to Numerical Analysis, Second edition, John Wiley & Sons, Inc., New York (1989). MR 90m:65001
 4.
A. Beurling and L.V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math., 96 (1956), 125142.MR 19:258c
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L. Carleson, On mappings, conformal at the boundary, J. Analyse Math., 19 (1967), 113. MR 35:6821
 6.
B.E.J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math., 108 (1986), 11191138. MR 88i:35061
 7.
E. Dyn'kin, Estimates for asymptotically conformal mappings, Ann. Acad. Sci. Fenn. Math., 22 (1997), no. 2, 275304. MR 98h:30030
 8.
R.A. Fefferman, C.E. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2), 134 (1991), 65124. MR 93h:31010
 9.
F.P. Gardiner and D.P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math., 114 (1992), 683736. MR 95h:30020
 10.
D.H. Hamilton, Rectifiable Julia curves, J. London Math. Soc. (2), 54 (1996), no. 3, 530540. MR 98j:30022
 11.
O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane. Second edition. SpringerVerlag, New YorkHeidelberg (1973).MR 49:9202
 12.
D.G. Moursund and C.S. Duris, Elementary Theory and Applications of Numerical Analysis, Dover Publications, Inc., New York (1988).MR 92a:65006
 13.
I.G. Nikolaev and S.Z. Shefel, Differential properties of mappings that are conformal at a point, Siberian Math. J., 27, 1, (1986), 106114.MR 87i:30037
 14.
S. Semmes, Quasiconformal mappings and chordarc curves, Trans. Amer. Math. Soc., 306, (1988), 233263. MR 89j:30029
 15.
C.E. Weil, The Peano derivative: what's known and what isn't, Real Anal. Exchange, 9, (1983/84), no. 2, 354365. MR 86c:26007
 1.
 J.M. Anderson, A. Cantón and J.L. Fernández, On smoothness of symmetric mappings, Complex Var. Theory Appl., 37 (1998), 161169.
 2.
 J.M. Anderson and A. Hinkkanen, Quasiconformal selfmappings with smooth boundary values, Bull. London Math. Soc., 26 (1994), 549556.MR 96a:30018
 3.
 K.E. Atkinson, An Introduction to Numerical Analysis, Second edition, John Wiley & Sons, Inc., New York (1989). MR 90m:65001
 4.
 A. Beurling and L.V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math., 96 (1956), 125142.MR 19:258c
 5.
 L. Carleson, On mappings, conformal at the boundary, J. Analyse Math., 19 (1967), 113. MR 35:6821
 6.
 B.E.J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math., 108 (1986), 11191138. MR 88i:35061
 7.
 E. Dyn'kin, Estimates for asymptotically conformal mappings, Ann. Acad. Sci. Fenn. Math., 22 (1997), no. 2, 275304. MR 98h:30030
 8.
 R.A. Fefferman, C.E. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2), 134 (1991), 65124. MR 93h:31010
 9.
 F.P. Gardiner and D.P. Sullivan, Symmetric structures on a closed curve, Amer. J. Math., 114 (1992), 683736. MR 95h:30020
 10.
 D.H. Hamilton, Rectifiable Julia curves, J. London Math. Soc. (2), 54 (1996), no. 3, 530540. MR 98j:30022
 11.
 O. Lehto and K.I. Virtanen, Quasiconformal Mappings in the Plane. Second edition. SpringerVerlag, New YorkHeidelberg (1973).MR 49:9202
 12.
 D.G. Moursund and C.S. Duris, Elementary Theory and Applications of Numerical Analysis, Dover Publications, Inc., New York (1988).MR 92a:65006
 13.
 I.G. Nikolaev and S.Z. Shefel, Differential properties of mappings that are conformal at a point, Siberian Math. J., 27, 1, (1986), 106114.MR 87i:30037
 14.
 S. Semmes, Quasiconformal mappings and chordarc curves, Trans. Amer. Math. Soc., 306, (1988), 233263. MR 89j:30029
 15.
 C.E. Weil, The Peano derivative: what's known and what isn't, Real Anal. Exchange, 9, (1983/84), no. 2, 354365. MR 86c:26007
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Additional Information
A. Cantón
Affiliation:
Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 981954350
Address at time of publication:
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Email:
acanton@mat.uab.es
DOI:
http://dx.doi.org/10.1090/S0002993904074982
PII:
S 00029939(04)074982
Keywords:
Quasiconformal mapping,
quasisymmetric mapping,
BeurlingAhlfors extension
Received by editor(s):
April 16, 2003
Received by editor(s) in revised form:
September 4, 2003
Published electronically:
June 2, 2004
Additional Notes:
The author’s research was supported by an FPI grant from Ministerio de Educación y Cultura (Spain) and a grant from MECD while visiting the University of Washington.
Communicated by:
Juha M. Heinonen
Article copyright:
© Copyright 2004 American Mathematical Society
