On a Sobolev-type inequality

Abstract

A new proof of the classical Sobolev inequality in $R^n$ with the best constant is given. The result follows from an intermediate inequality which connects in a sharp way the $L^p$ norm of the gradient of a function $u$ to $L^{p^{\ast }}$ and $L^{p^{\ast }}$-weak norms of $u$, where $p\in ]1;n[$ and $p^{\ast }=\frac{np}{n-p}$ is the Sobolev exponent.