Symmetric Decreasing Rearrangement Is Sometimes Continuous

This paper deals with the operation <!-- MATH $\mathcal{R}$ --> of symmetric decreasing rearrangement which maps <!-- MATH ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$ --> to <!-- MATH ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$ --> . We show that even though it is norm decreasing, <!-- MATH $\mathcal{R}$ --> is not continuous for . The functions at which <!-- MATH $\mathcal{R}$ --> 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 <!-- MATH ${{\mathbf{W}}^{1,p}}({{\mathbf{R}}^n})$ --> . Curiously, <!-- MATH $\mathcal{R}$ --> is always continuous in fractional Sobolev spaces <!-- MATH ${{\mathbf{W}}^{\alpha ,p}}({{\mathbf{R}}^n})$ --> with <!-- MATH $0 < \alpha < 1$ --> <img width="92" height="37" align="MIDDLE" border="0" src="images/img11.gif" alt="$ 0 < \alpha < 1$">.