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.