On a Sobolev-type inequality (Q1039878)

Summary: A new proof of the classical Sobolev inequality in \(\mathbb R^n\) 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 \(L^{p^*}\) and \(L^{p^*}\)-weak norms of \(u\), where \(p\in ]1, n[\) and \(p^* = \frac {np}{(n-p)}\) is the Sobolev exponent.