Symmetric decreasing rearrangement is sometimes continuous

@article{Almgren1989SymmetricDR,
  title={Symmetric decreasing rearrangement is sometimes continuous},
  author={Frederick J. Almgren and Elliott H. Lieb},
  journal={Journal of the American Mathematical Society},
  year={1989},
  volume={2},
  pages={683-773}
}
  • F. Almgren, E. Lieb
  • Published 13 January 1989
  • Mathematics
  • Journal of the American Mathematical Society
This paper deals with the operation .9R 'of symmetric decreasing rearrangement which maps Wl,p (Rn) to Wi ,p (Rn) . We show that even though it is norm decreasing, .9R is not continuous for n ~ 2. The functions at which .9R is continuous are precisely characterized by a new property called co-area regularity. Every sufficiently differentiable function is co-area regular, and both the regular and the irregular functions are dense in Wi ,P(Rn ). Curiously, .9R is always continuous in fractional… 

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