Skip Navigation

International Mathematics Research Notices (2008) Vol. 2008 : article ID rnn047, 43 pages, doi:10.1093/imrn/rnn047 published on June 11, 2008
This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Uriarte-Tuero, I.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?

© The Author 2008. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permissions@oxfordjournals.org

Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability

Ignacio Uriarte-Tuero

Mathematics Department, 202 Mathematical Sciences Bldg., University of Missouri, Columbia, MO 65211-4100, USA and Fields Institute, 222 College Street, Toronto, Ontario M5T 3J1, Canada

Correspondence: Correspondence to be sent to: ignacio{at}math.missouri.edu

In the celebrated paper [5, Acta Mathematica, 173, 1994], Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in the joint work of Astala, Clop, Mateu, Orobitg and the author [6, Duke Mathematical Journal, forthcoming] (Theorem 1.1), the other cases remaining open. A related question that we left open in [6] (Question 4.2) (which was asked by Astala to the author before [6] in an equivalent form [4, Personal communication]) is whether BMO removability for K-quasiregular mappings and (L{infty}) removability for K-quasiregular mappings are indeed different problems. In this paper, we give a series of examples answering the positive Question 4.2 in [6], at the same time proving sharpness in two different senses of Theorem 1.1 in [6], and also giving examples that would yield sharpness in those two different senses as well for the absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings, were it to be proved.



References

  1. Adams D. R., Hedberg L. I. Function Spaces and Potential Theory (1996) Berlin: Springer. Fundamental Principles of Mathematical Sciences 314.
  2. Ahlfors L. V. Bounded analytic functions. Duke Mathematical Journal (1947) 14:1–11.[CrossRef][ISI]
  3. Ahlfors L. V. Lectures on Quasiconformal Mappings (1966) New York: D. Van Nostrand Co. Mathematical Studies 10.
  4. Astala K. Personal communication.
  5. Astala K. Area distortion of quasiconformal mappings. Acta Mathematica (1994) 173(1):37–60.[CrossRef][ISI]
  6. Astala K., Clop A., Mateu J., Orobitg J., Uriarte-Tuero I. Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings. Duke Mathematical Journal (2008) 141(3):539–71.[CrossRef][ISI]
  7. Astala K., Iwaniec T., Martin G. Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. (forthcoming).
  8. Astala K., Iwaniec T., Saksman E. Beltrami operators in the plane. Duke Mathematical Journal (2001) 107(1):27–56.[CrossRef][ISI]
  9. Bishop C. J. Distortion of disks by conformal maps. Preprint.
  10. Calderón A.-P. Cauchy integrals on Lipschitz curves and related operators. Proceedings of the National Academy of Sciences of the United States of America (1977) 74(4):1324–7.[Abstract/Free Full Text]
  11. Clop A. Removable singularities for planar quasiregular mappings. (2006) Ph.D. Dissertation, Universitat Autonoma de Barcelona.
  12. Clop A. Nonremovable sets for Hölder continuous quasiregular mappings in the plane. The Michigan Mathematical Journal (2007) 55(1):195–208.[CrossRef]
  13. Clop A. Removable singularities for Hölder continuous quasiregular mappings in the plane. Annales Academi Scientiarium Fennicae Mathematica (2007) 32(1):171–8.
  14. David G. Unrectifiable 1-sets have vanishing analytic capacity. Revista Matemática Iberoamericana (1998) 14(2):369–479.[ISI]
  15. Garabedian P. R. Schwarz's lemma and the Szegö kernel function. Transactions of the American Mathematical Society (1949) 67:1–35.[CrossRef][ISI]
  16. Iwaniec T., Martin G. Quasiregular mappings in even dimensions. Acta Mathematica (1993) 170(1):29–81.[CrossRef][ISI]
  17. Iwaniec T., Martin G. Geometric Function Theory and Non-Linear Analysis (2001) Oxford Mathematical Monographs. New York: Oxford University Press.
  18. Kaufman R. Hausdorff measure, BMO, and analytic functions. Pacific Journal of Mathematics (1982) 102(2):369–71.[ISI]
  19. Král J. Semielliptic singularities. Casopis Pest. Mat. (1984) 109(3):304–22.
  20. Lacey M., Sawyer E., Uriarte-Tuero I. Astala's conjecture on distortion of Hausdorff measures under quasiconformal maps in the plane. (2008) preprint arXiv:0805.4711v1.
  21. Lehto O., Virtanen K. I. Quasiconformal Mappings in the Plane. Translated by Lucas K. W. (1973) 2nd ed. New York: Springer.
  22. Mattila P. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability (1995) Cambridge, United Kingdom: Cambridge University Press. Cambridge Studies in Advanced Mathematics 44.
  23. Mori A. On an absolute constant in the theory of quasi-conformal mappings. Journal of the Mathematical Society of Japan (1956) 8:156–66.
  24. Rogers C. A. Hausdorff Measures (1998) Cambridge Mathematical Library. Cambridge, United Kingdom: Cambridge University Press. Reprint of the 1970 original, with a foreword by K. J. Falconer.
  25. Tolsa X. Painlevé's problem and the semiadditivity of analytic capacity. Acta Mathematica (2003) 190(1):105–49.[CrossRef][ISI]
  26. Tolsa X. Bilipschitz maps, analytic capacity, and the Cauchy integral. Annals of Mathematics (2005) 162(3):1243–304.[ISI]
  27. Willard S. General Topology (1970) Reading, MA: Addison-Wesley.

Add to CiteULike CiteULike   Add to Connotea Connotea   Add to Del.icio.us Del.icio.us    What's this?



This Article
Right arrow Abstract Freely available
Right arrow Full Text (PDF)
Right arrow Alert me when this article is cited
Right arrow Alert me if a correction is posted
Services
Right arrow Email this article to a friend
Right arrow Similar articles in this journal
Right arrow Alert me to new issues of the journal
Right arrow Add to My Personal Archive
Right arrow Download to citation manager
Right arrowRequest Permissions
Right arrow How to cite this article
Google Scholar
Right arrow Articles by Uriarte-Tuero, I.
Social Bookmarking
Add to CiteULike   Add to Connotea   Add to Del.icio.us  
What's this?