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 Lp norm of the gradient of a function u to L__p* and L__p*-weak norms of u, where p ∈ ]1; n[ and p* = np/(n-p) is the Sobolev exponent.