WIAS Preprint No. 1949, (2014)

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


  • Ferreira, Rita
  • Kreisbeck, Carolin
  • Ribeiro, Ana Margarida

2010 Mathematics Subject Classification

  • 46E35


  • Higher-order Sobolev spaces, nonlocal functionals




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 [Russ. Math. Surv. 57 (2002), pp. 693-708] to the higher-order case, and extend the work by Borghol [Asymptotic Anal. 51 (2007), pp. 303-318] to a more general setting.

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