Generalized dimension compression under mappings of exponentially integrable distortion

Aleksandra Zapadinskaya

Open Mathematics (2011)

  • Volume: 9, Issue: 2, page 356-363
  • ISSN: 2391-5455

Abstract

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We prove a dimension compression estimate for homeomorphic mappings of exponentially integrable distortion via a modulus of continuity result by D. Herron and P. Koskela [Mappings of finite distortion: gauge dimension of generalized quasicircles, Illinois J. Math., 2003, 47(4), 1243–1259]. The essential sharpness of our estimate is demonstrated by an example.

How to cite

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Aleksandra Zapadinskaya. "Generalized dimension compression under mappings of exponentially integrable distortion." Open Mathematics 9.2 (2011): 356-363. <http://eudml.org/doc/269401>.

@article{AleksandraZapadinskaya2011,
abstract = {We prove a dimension compression estimate for homeomorphic mappings of exponentially integrable distortion via a modulus of continuity result by D. Herron and P. Koskela [Mappings of finite distortion: gauge dimension of generalized quasicircles, Illinois J. Math., 2003, 47(4), 1243–1259]. The essential sharpness of our estimate is demonstrated by an example.},
author = {Aleksandra Zapadinskaya},
journal = {Open Mathematics},
keywords = {Mapping of finite distortion; Exponentially integrable distortion; Generalized Hausdorff measure; Hausdorff dimension; mapping of finite distortion; exponentially integrable distortion; generalized Hausdorff measure},
language = {eng},
number = {2},
pages = {356-363},
title = {Generalized dimension compression under mappings of exponentially integrable distortion},
url = {http://eudml.org/doc/269401},
volume = {9},
year = {2011},
}

TY - JOUR
AU - Aleksandra Zapadinskaya
TI - Generalized dimension compression under mappings of exponentially integrable distortion
JO - Open Mathematics
PY - 2011
VL - 9
IS - 2
SP - 356
EP - 363
AB - We prove a dimension compression estimate for homeomorphic mappings of exponentially integrable distortion via a modulus of continuity result by D. Herron and P. Koskela [Mappings of finite distortion: gauge dimension of generalized quasicircles, Illinois J. Math., 2003, 47(4), 1243–1259]. The essential sharpness of our estimate is demonstrated by an example.
LA - eng
KW - Mapping of finite distortion; Exponentially integrable distortion; Generalized Hausdorff measure; Hausdorff dimension; mapping of finite distortion; exponentially integrable distortion; generalized Hausdorff measure
UR - http://eudml.org/doc/269401
ER -

References

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  1. [1] Astala K., Gill J.T., Rohde S., Saksman E., Optimal regularity for planar mappings of finite distortion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2010, 27(1), 1–19 http://dx.doi.org/10.1016/j.anihpc.2009.01.012 Zbl1191.30007
  2. [2] Astala K., Iwaniec T., Koskela P., Martin G., Mappings of BMO-bounded distortion, Math. Ann., 2000, 317(4), 703–726 http://dx.doi.org/10.1007/PL00004420 Zbl0954.30009
  3. [3] Astala K., Iwaniec T., Martin G., Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton Math. Ser., 48, Princeton University Press, Princeton, 2009 Zbl1182.30001
  4. [4] Boyarskiĭ B.V., Homeomorphic solutions of Beltrami systems, Dokl. Akad. Nauk SSSR (N.S.), 1955, 102, 661–664 
  5. [5] David G., Solutions de l’équation de Beltrami avec ‖µ‖∞ = 1, Ann. Acad. Sci. Fenn. Ser. A I Math., 1988, 13(1), 25–70 Zbl0619.30024
  6. [6] Falconer K., Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, Chichester, 1990 
  7. [7] Faraco D., Koskela P., Zhong X., Mappings of finite distortion: the degree of regularity, Adv. Math., 2005, 190(2), 300–318 http://dx.doi.org/10.1016/j.aim.2003.12.009 Zbl1075.30012
  8. [8] Gehring F.W., The L p-integrability of the partial derivatives of a quasiconformal mapping, Acta Math., 1973, 130(1), 265–277 http://dx.doi.org/10.1007/BF02392268 Zbl0258.30021
  9. [9] Herron D.A., Koskela P., Mappings of finite distortion: gauge dimension of generalized quasicircles, Illinois J. Math., 2003, 47(4), 1243–1259 Zbl1050.30012
  10. [10] Koskela P., Zapadinskaya A., Zürcher T., Mappings of finite distortion: generalized Hausdorff dimension distortion, J. Geom. Anal., 2010, 20(3), 690–704 http://dx.doi.org/10.1007/s12220-010-9121-8 Zbl1205.30017
  11. [11] Rajala T., Zapadinskaya A., Zürcher T., Generalized Hausdorff dimension distortion in Euclidean spaces under Sobolev mappings, preprint available at http://arxiv.org/abs/1007.2091 Zbl1252.46024

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