Characterizations of second order Sobolev spaces

Second order Sobolev spaces are important in applications to partial differential equations and geometric analysis, in particular to equations such as the bi-Laplacian. The main purpose of this paper is to establish some new characterizations of the second order Sobolev spaces W2,p(RN)W2,p(RN) in Euclidean spaces. We will present here several types of characterizations: by second order differences, by the Taylor remainder of first order and by the differences of the first order gradient. Such characterizations are inspired by the works of Bourgain et al. (2001) and Nguyen (2006, 2008) on characterizations of first order Sobolev spaces in the Euclidean space.