Symmetric decreasing rearrangement is sometimes continuous

Abstract

This paper deals with the operation R \mathcal {R} of symmetric decreasing rearrangement which maps W 1 , p ( R n ) {{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n}) to W 1 , p ( R n ) {{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n}) . We show that even though it is norm decreasing, R \mathcal {R} is not continuous for n ≥ 2 n \geq 2 . The functions at which R \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 W 1 , p ( R n ) {{\mathbf {W}}^{1,p}}({{\mathbf {R}}^n}) . Curiously, R \mathcal {R} is always continuous in fractional Sobolev spaces W α , p ( R n ) {{\mathbf {W}}^{\alpha ,p}}({{\mathbf {R}}^n}) with 0 > α > 1 0 > \alpha > 1 .