Sharp Poincaré Inequalities for Families of Nonlocal Nonconvex Functionals

Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincaré inequalities for the nonlocal nonconvex functionals $$ Φ_{λ,p}^γ(g;Q_0) :=λ^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>λ|x-y|^{1+\fracγ{p}}\}} |x-y|^{γ-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R