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Symmetric decreasing rearrangement is sometimes continuous

Authors: Frederick J. Almgren and Elliott H. Lieb
Journal: J. Amer. Math. Soc. 2 (1989), 683-773
MSC: Primary 49F20; Secondary 46E30, 49A50
MathSciNet review: 1002633
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Abstract: This paper deals with the operation $ \mathcal{R}$ of symmetric decreasing rearrangement which maps $ {{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$ to $ {{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$. We show that even though it is norm decreasing, $ \mathcal{R}$ is not continuous for $ n \geq 2$. The functions at which $ \mathcal{R}$ 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 $ {{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$. Curiously, $ \mathcal{R}$ is always continuous in fractional Sobolev spaces $ {{\mathbf{W}}^{\alpha ,p}}({{\mathbf{R}}^n})$ with $ 0 < \alpha < 1$.

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