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

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 (Omega), k N and p (1,+∞), for arbitrary open sets Omega 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.