A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities.
The authors give a new proof of the sharp Sobolev embedding in \(R^n\), \[ \left( \int_{R^n} | f(x)| ^{p^*} \, dx \right)^{1/p^*} \leq S_n (p) \, \left( \int_{R^n} | \nabla f| ^p \, dx \right)^{1/p} \] where \[ 1<p<n, \quad 1- n/p = - n/p^*. \] The sharp constants \(S_n (p)\) and related extremal functions are calculated using new measure-theoretical arguments, called mass-transportation. Similar results are obtained for \(p=1\) (related to isoperimetrical problems). Euclidean norms are replaced by more general norms. Some results for related Gagliardo-Nirenberg inequalities are obtained.
scientific article; zbMATH DE number 2055356