Symmetric decreasing rearrangement is sometimes continuous

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 $0 < \alpha < 1$.