Families of functionals representing Sobolev norms

We obtain new characterizations of the Sobolev spaces Ẇ (R ) and the bounded variation space ̇ BV (R ). The characterizations are in terms of the functionals νγ(Eλ,γ/p[u]) where Eλ,γ/p[u] = { (x, y) ∈ R × R : x 6= y, |u(x) − u(y)| |x − y|1+γ/p > λ } and the measure νγ is given by dνγ(x, y) = |x−y| dxdy. We provide characterizations which involve the L-quasi-norms supλ>0 λ νγ(Eλ,γ/p[u]) 1/p and also exact formulas via corresponding limit functionals, with the limit for λ → ∞ when γ > 0 and the limit for λ → 0 when γ < 0. The results unify and substantially extend previous work by Nguyen and by Brezis, Van Schaftingen and Yung. For p > 1 the characterizations hold for all γ 6= 0. For p = 1 the upper bounds for the L quasi-norms fail in the range γ ∈ [−1, 0); moreover in this case the limit functionals represent the L norm of the gradient for C c -functions but not for generic Ẇ -functions. For this situation we provide new counterexamples which are built on self-similar sets of dimension γ + 1. For γ = 0 the characterizations of Sobolev spaces fail; however we obtain a new formula for the Lipschitz norm via the expressions ν0(Eλ,0[u]).