Sharp Truncated Difference Estimates for Nonlocal Nonconvex Functionals Defined by Superlevel Sets

Abstract

Let $N\in \mathbb{N}$ and $p\in [1,\infty )$. In this article, we consider a family $\{{\Phi }_{\lambda ,p}^{\gamma }{\}}_{\lambda \in (0,\infty )}$ of nonlocal nonconvex functionals with a real parameter $\gamma$ and establish two truncated difference estimates bounded by these functionals with the sharp range of $\gamma$; that is, for any $\gamma \in (-\infty ,-1]$, any $\lambda \in (0,\infty )$, any cube $Q\subset {\mathbb{R}}^N$, and any measurable function $g:Q\to \mathbb{R}$, \begin{align*} &\int_Q\int_Q\left[\frac{|g(x)-g(y)|-λ|x-y|^{1+γ/p}}{\ell(Q)^{1+γ/p}}\right]_+^p\,dx\,dy\\ &\quad\lesssim λ^p\mathop{\int_Q\int_Q}\limits_{\frac{|g(x)-g(y)|}{|x-y|^{1+γ/p}}>λ}\left[\frac{|x-y|}{\ell(Q)}\right]^{γ-N}\,dx\,dy=:\frac{Φ_{λ,p}^γ(g;Q)}{\ell(Q)^{γ-N}}, \end{align*} where the implicit positive constant depends only on $N$, $p$, and $\gamma$. For $\gamma \in (0,\infty )$, the above estimate holds with the left-most integrand replaced by $\frac{\min \{|g(x)-g(y){|}^p,{\lambda }^p|x-y{|}^{p+\gamma }\}}{\ell (Q{)}^{p+\gamma }}$. As applications, these estimates yield related nonlocal Poincaré inequalities with the sharp range of $\gamma$, and also recover the known $\Gamma$-liminf lower bounds and characterizations of Sobolev and BV spaces in terms of the liminf of the functionals $\{{\Phi }_{\lambda ,p}^{\gamma }{\}}_{\lambda \in (0,\infty )}$ for all $\gamma \in \mathbb{R}\setminus (-1,0]$. We investigate further applications to compactness and variational problems as well as a new VMO criterion. Inspired by a recent work of R. L. Frank, we develop an elementary approach based on dyadic iteration of truncated differences along line segments, which allows us to treat the aforementioned problems in a unified way.

arXiv commentA new elementary argument unifies and simplifies the proofs in the original version