Families of functionals representing Sobolev norms

We obtain new characterizations of the Sobolev spaces ${Ẇ}^{1,p}({ℝ}^N)$ and the bounded variation space $\dot{\mathrm{BV}<!--FUNCTION\; APPLICATION-->}({ℝ}^N)$. The characterizations are in terms of the functionals ${\nu }_{\gamma }(E_{\lambda ,\gamma ∕p}[u])$, where

$$E_{\lambda ,\gamma ∕p}[u]=\left\{(x,y)\in {ℝ}^N\times {ℝ}^N:x\ne y,\frac{|u(x)-u(y)|}{|x-y{|}^{1+\gamma ∕p}}>\lambda \right\}$$

and the measure ${\nu }_{\gamma }$ is given by $d<!--FUNCTION\; APPLICATION-->{\nu }_{\gamma }(x,y)=|x-y{|}^{\gamma -N}d<!--FUNCTION\; APPLICATION-->xd<!--FUNCTION\; APPLICATION-->y$. We provide characterizations which involve the $L^{p,\infty }$-quasinorms ${\sup <!--FUNCTION\; APPLICATION-->}_{\lambda >0}\lambda {\nu }_{\gamma }{(E_{\lambda ,\gamma ∕p}[u])}^{1∕p}$ and also exact formulas via corresponding limit functionals, with the limit for $\lambda \to \infty$ when $\gamma >0$ and the limit for $\lambda \to 0^{+}$ when $\gamma <0$. 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 $\gamma \ne 0$. For $p=1$ the upper bounds for the $L^{1,\infty }$ quasinorms fail in the range $\gamma \in [-1,0)$; moreover, in this case the limit functionals represent the $L^1$ norm of the gradient for $C_c^{\infty }$-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 $\gamma +1$. For $\gamma =0$ the characterizations of Sobolev spaces fail; however, we obtain a new formula for the Lipschitz norm via the expressions ${\nu }_0(E_{\lambda ,0}[u])$.