Families of functionals representing Sobolev norms

We obtain new characterizations of the Sobolev spaces Ẇ 1, p ‫ޒ(‬ N ) and the bounded variation space ḂV(‫ޒ‬ N ).The characterizations are in terms of the functionals ν γ (E λ,γ / p [u]), whereand the measure ν γ is given by dν γ (x, y) = |x -y| γ -N dx dy.We provide characterizations which involve the L p,∞ -quasinorms sup λ>0 λν γ (E λ,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 L 1,∞ quasinorms fail in the range γ ∈ [-1, 0); moreover, in this case the limit functionals represent the L 1 norm of the gradient for C ∞ c -functions but not for generic Ẇ 1,1 -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 ν 0 (E λ,0 [u]).