On the (non)compactness of the radial Sobolev spaces
The authors establish compactness and noncompactness results to the embedding operators from Sobolev spaces of functions in radial form to \(L^ p\)-spaces: We show that (i) \(H^ m_ r({\mathbb{R}}^ n)\hookrightarrow L^ p({\mathbb{R}}^ n)\) \((2<p<\frac{2n}{n-2m}\), \(n<2m)\) is compact (ii) \(H^ m_ r({\mathbb{R}}^ n)\hookrightarrow L^ p({\mathbb{R}}^ n)\) \((2<p<\infty\), \(n<2m)\) is compact (iii) for any \(m\in N\), \(H^ m_ r({\mathbb{R}}^ n)\hookrightarrow L^ 2({\mathbb{R}}^ n)\) is not compact, (iv) \(H^ m_ r({\mathbb{R}}^ n)\hookrightarrow L^ q({\mathbb{R}}^ n)\) \((q=\frac{2n}{n-2m},n>2m)\) is not compact.
Sobolev spaces of functions in radial form
scientific article; zbMATH DE number 4000746