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The affine Sobolev inequality. (English) Zbl 1040.53089

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.

MSC:

53C65 Integral geometry
52A39 Mixed volumes and related topics in convex geometry
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
26D10 Inequalities involving derivatives and differential and integral operators
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