Some properties of Sobolev spaces

The aim of this paper is to present some new characterizations of the Sobolev spaces W^k,p(Ω) where 1<p<∞, k≥2 and Ω is a bounded smooth domain of $\mathbb{R}$ ^N. For p=1, we prove same results by replacing W^k,1(Ω) by the BV^k(Ω) space. As an application we show how to recognize polynomial functions among locally integrable functions.