Families of Functionals Representing Sobolev Norms

We obtain new characterizations of the Sobolev spaces Ẇ1,p(RN) and the bounded variation space ˙BV(RN). The characterizations are in terms of the functionals νγ(Eλ,γ∕p[u]), where

Eλ,γ∕p[u]={(x,y)∈RN×RN:x≠y,|u(x)−u(y)|∣∣x−y∣∣1+γ∕p>λ}

and the measure νγ is given by dνγ(x,y)=∣∣x−y∣∣γ−Ndxdy. We provide characterizations which involve the Lp,∞-quasinorms supλ>0λνγ(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≠0. For p=1 the upper bounds for the L1,∞ quasinorms fail in the range γ∈[−1,0); moreover, in this case the limit functionals represent the L1 norm of the gradient for C∞c-functions but not for generic Ẇ1,1-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]).