Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients

The aim of this paper, which deals with a class of singular functionals involving difference quotients, is twofold: deriving suitable integral conditions under which a measurable function is polynomial and stating necessary and sufficient criteria for an integrable function to belong to a kkth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω)Wk,p(Ω), k∈Nk∈N and p∈(1,+∞)p∈(1,+∞), for arbitrary open sets Ω⊂RnΩ⊂Rn. In particular, we provide natural generalizations of the results regarding Sobolev spaces summarized in Brézis’ overview article [Brézis (2002)] to the higher-order case, and extend the work [Borghol (2007)] to a more general setting.