Illinois Journal of Mathematics

Hölder continuous Sobolev mappings and the Lusin N property

Aleksandra Zapadinskaya

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We give a new proof for the result of J. Malý and O. Martio, stating that Hölder continuous mappings in $W^{1,n}$ satisfy the Lusin N property. We further generalize this result to a metric setting.

Article information

Illinois J. Math., Volume 58, Number 2 (2014), 585-591.

Received: 18 June 2014
Revised: 2 July 2014
First available in Project Euclid: 7 July 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46E35: Sobolev spaces and other spaces of "smooth" functions, embedding theorems, trace theorems 26B35: Special properties of functions of several variables, Hölder conditions, etc. 28A75: Length, area, volume, other geometric measure theory [See also 26B15, 49Q15] 28A78: Hausdorff and packing measures


Zapadinskaya, Aleksandra. Hölder continuous Sobolev mappings and the Lusin N property. Illinois J. Math. 58 (2014), no. 2, 585--591. doi:10.1215/ijm/1436275500.

Export citation


  • J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer, New York, 2001.
  • J. Heinonen, P. Koskela, N. Shanmugalingam and J. T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87–139.
  • J. D. Howroyd, On the theory of Hausdorff measures in metric spaces, Ph.D. thesis, University College London, 1994; available at
  • A. Kauranen and P. Koskela, Boundary blow up under Sobolev mappings, Anal. PDE 7 (2014), 1839–1850.
  • P. Koskela, J. Malý and T. Zürcher, Lusin's condition N and modulus of continuity, Adv. Calc. Var. 8 (2015), 155–171.
  • P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Math. Ann. 309 (1997), no. 4, 593–609.
  • J. Malý, The area formula for $W^{1,n}$-mappings, Comment. Math. Univ. Carolin. 35 (1994), no. 2, 291–298.
  • J. Malý and O. Martio, Lusin's condition (N) and mappings of the class $W^{1,n}$, J. Reine Angew. Math. 458 (1995), 19–36.
  • M. Marcus and V. J. Mizel, Transformations by functions in Sobolev spaces and lower semicontinuity for parametric variational problems, Bull. Amer. Math. Soc. (N.S.) 79 (1973), 790–795.
  • O. Martio and W. P. Ziemer, Lusin's condition (N) and mappings with nonnegative Jacobians, Michigan Math. J. 39 (1992), no. 3, 495–508.
  • J. G. Rešetnjak, Certain geometric properties of functions and mappings with generalized derivatives, Sibirsk. Mat. \u Z. 7 (1966), 886–919.
  • Y. G. Reshetnyak, The $N$ condition for spatial mappings of the class $W^1_{n,{\rm loc}}$, Sibirsk. Mat. Zh. 28 (1987), no. 5, 149–153.