Abstract
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 kth-order Sobolev space. One of the main theorems is a new characterization of Wk,p(Ω), k∈N and p∈(1,+∞), for arbitrary open sets Ω⊂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.
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