# 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} }

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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