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A note on the support of a Sobolev function on a $k$-cell

Author: W. K. Ziemer
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1987-1995
MSC (2000): Primary 46E35
Published electronically: December 19, 2003
MathSciNet review: 2053970
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Abstract: It is shown that a $k$-cell (the homeomorphic image of a closed ball in $\mathbb{R} ^{k}$) in $\mathbb{R} ^{n}$, $1\leq k<n$, cannot support a function in $W^{1,p}(\mathbb{R} ^{n})$ if $p>[\frac{k+1}{2}]$, the greatest integer in $(k+1)/2$.

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  • [AH96] D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer-Verlag, Berlin, 1996. MR 97j:46024
  • [Bes50] A. Besicovitch, Parametric Surfaces, Bulletin of the American Mathematical Society, 56 (1950), pp. 288-296. MR 12:168e
  • [BG98] T. Bagby and P.M. Gauthier, Note on the support of Sobolev functions, Canadian Math. Bulletin 41 (3) (1998), pp. 257-260. MR 99g:46037
  • [Gage81] M. Gage, On Gehring's Linked Sphere Problem, American Journal of Mathematics, 103, no. 3. (June, 1981), pp. 437-443. MR 82j:52024
  • [Fed59] H. Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (3) (1959), pp. 418-491. MR 22:961
  • [GT83] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Second Edition, Springer-Verlag, 1983. MR 86c:35035
  • [Hir76] M. W. Hirsch, Differential Topology, Springer-Verlag, New York, 1976. MR 56:6669
  • [MZ97] Jan Malý and W. P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, Math. Surveys and Monographs, Amer. Math. Soc., Providence, RI, 1997. MR 98h:35080
  • [Mas91] W. S. Massey, A Basic Course in Algebraic Topology, Springer-Verlag, New York, 1991. MR 92c:55001
  • [Pol72] J. C. Polking, A Leibnitz formula for some differentiation operators of fractional order, Indiana Univ. Math. J. 21 (1972) pp. 1019-1029. MR 47:7414
  • [Pon87] S. P. Ponomarev, On the $N$-property of homeomorphisms of the class $\mathit{W}_{p}^{1}$, Siberian Math. Journal 28 (1987) pp. 291-298. MR 88i:26031
  • [Rol76] D. Rolfson, Knots and Links, Publish or Perish, Berkeley, California, 1976. MR 58:24236
  • [Zie89] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. MR 91e:46046

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Additional Information

W. K. Ziemer
Affiliation: Department of Mathematics, California State University Long Beach, Long Beach, California 90840-1001

Received by editor(s): May 3, 2001
Received by editor(s) in revised form: March 11, 2003
Published electronically: December 19, 2003
Communicated by: David Preiss
Article copyright: © Copyright 2003 American Mathematical Society

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