The purpose of this paper is to provide a new characterization of the Sobolev space W 1,1(Rn). We also show a new proof of the characterization of the Sobolev space W 1,p(Rn), 1 ≤ p <∞, in terms of Poincaré inequalities. The Sobolev space W 1,p(Rn), 1 ≤ p < ∞, consists of functions u ∈ Lp(Rn) such that |∇u| ∈ Lp(Rn). It is a Banach space with respect to the… (More)
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