The affine Sobolev inequality.
The author proves a new Sobolev inequality which is stronger than its classical, Euclidean counterpart. The main theorem in the article states, in fact, that if \(f\) is a \(C^1\) function with compact support in \(\mathbb{R}^n\), then \[ {1\over n} \int_{S^{n-1}} \|\nabla_u f\|^{-n}_1du\leq c_n\| f\|^{-n}_{{n\over n-1}}, \] where \(\nabla_u f\) is the partial derivative of \(f\) in direction \(u\), \(du\) is the standard surface measure on the unit sphere, and the constant \(c_n= ({\omega_n\over 2\omega_{n-1}})^n\) is best. In the latter expression \(\omega_n\) denotes the volume enclosed by the unit sphere \(S^{n-1}\) in \(\mathbb{R}^n\), and \(\| f\|_p\) is the usual \(L_p\) norm of \(f\) in \(\mathbb{R}^n\). This inequality is invariant under the action of \(\text{GL}(n)\) and, as a consequence, does not depend on the Euclidean norm of \(\mathbb{R}^n\). The best constant is attained at the characteristic functions of ellipsoids. A generalization of the Gagliardo-Nirenberg inequality is also proven in the paper.
scientific article; zbMATH DE number 1782621