Families of functionals representing Sobolev norms

Abstract

We obtain new characterizations of the Sobolev spaces ${\dot{W}}^{1,p}({\mathbb{R}}^N)$ and the bounded variation space $\dot{BV}({\mathbb{R}}^N)$. The characterizations are in terms of the functionals ${\nu }_{\gamma }(E_{\lambda ,\gamma /p}[u])$ where \[ E_{λ,γ/p}[u]= \Big\{(x,y )\in \mathbb{R}^N \times \mathbb{R}^N \colon x \neq y, \, \frac{|u(x)-u(y)|}{|x-y|^{1+γ/p}}>λ\Big\} \] and the measure ${\nu }_{\gamma }$ is given by $\mathrm{d}{\nu }_{\gamma }(x,y)=|x-y{|}^{\gamma -N}\mathrm{d}x\mathrm{d}y$. We provide characterizations which involve the $L^{p,\infty }$-quasi-norms ${\sup }_{\lambda >0}\lambda {\nu }_{\gamma }(E_{\lambda ,\gamma /p}[u]{)}^{1/p}$ 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 $L^{1,\infty }$ quasi-norms 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 ${\dot{W}}^{1,1}$-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 ${\nu }_0(E_{\lambda ,0}[u])$.

40 pages

References in corpus (2)

• A surprising formula for Sobolev norms
• Spaces of Besov-Sobolev type and a problem on nonlinear approximation

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