Illinois Journal of Mathematics

Peano cubes with derivatives in a Lorentz space

K. Wildrick and T. Zürcher

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We show that for any length-compact metric space $Y$ and any $1 \lt q\leq n$, there is a continuous surjection in a suitably defined Sobolev–Lorentz space space $W^{1,n,q}([0,1]^n, Y)$. On the other hand, we show that mappings in the space $W^{1,n,1}([0,1]^n, Y)$ satisfy condition (N). This implies that the target $Y$ can be at most $n$-dimensional.

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Illinois J. Math., Volume 53, Number 2 (2009), 365-378.

First available in Project Euclid: 23 February 2010

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Primary: 26B35: Special properties of functions of several variables, Hölder conditions, etc.


Wildrick, K.; Zürcher, T. Peano cubes with derivatives in a Lorentz space. Illinois J. Math. 53 (2009), no. 2, 365--378. doi:10.1215/ijm/1266934782.

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  • C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics, vol. 129, Academic Press, Boston, MA, 1988.
  • P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), x–101.
  • P. Hajłasz and J. T. Tyson, Sobolev Peano cubes, Michigan Math. J. 56 687–702.
  • J. Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, 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. Kauhanen, P. Koskela and J. Malý, On functions with derivatives in a Lorentz space, Manuscripta Math. 100 (1999), 87–101.
  • B. Kirchheim, Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), 113–123.
  • Y. G. Reshetnyak, Sobolev classes of functions with values in a metric space, Sibirsk. Mat. Zh. 38 (1997), 657–675.
  • E. M. Stein, Editor's note: The differentiability of functions in ${\bf R}\sp{n}$, Ann. of Math. (2) 113 (1981), 383–385.
  • W. P. Ziemer, Weakly differentiable functions, Graduate Texts in Mathematics, vol. 120, Springer-Verlag, New York, 1989.