A BMO-Type Characterization of Higher Order Sobolev Spaces

We obtain a new characterization of the higher Sobolev space $W^{m,p}(\mathbb R^{n})$ W m , p ( ℝ n ) , $m\in \mathbb N$ m ∈ ℕ and $p\in (1, +\infty )$ p ∈ ( 1 , + ∞ ) and of the space BVm, the space of functions of higher order bounded variation. The characterizations are in term of BMO-type seminorms. The results unify and substantially extend previous results in Fusco et al. (ESAIM Control Optim. Calc Var., 24(2), 835–847 2018) and Farroni et al. (J. Funct. Anal., 278(9), 108451 2020).