Characterization of polynomials and higher-order Sobolev spaces in terms of functionals involving difference quotients. (January 2015) - Ferreira, Rita; Kreisbeck, Carolin; Ribeiro, Ana Margarida
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 k th-order Sobolev space. One of the main theorems is a new characterization of W k, p ( Ω ), k ∈ N and p ∈ ( 1, + ∞ ), for arbitrary open sets Ω ⊂ R n . 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.