Sharp Nonlocal Poincaré Inequalities and Liminf Lower Bounds for Sobolev and BV Energies on Metric Measure Spaces

Abstract

Let $p\in [1,\infty )$, and let $(X,\rho ,\mu )$ be a complete doubling $q$-Poincaré space for some $q\in [1,p]$ with a lower dimension $d$ and an upper dimension $D$. In this article, we establish Poincaré inequalities for two families of nonlocal nonconvex functionals defined using distances or volumes. More precisely, for any ball $B$ with its radius $r_B\in (0,\mathrm{diam}X)$, any $\lambda \in (0,\infty )$, and any measurable function $f:X\to \mathbb{R}$, \begin{align*} &\frac{1}{μ(B)^2}\int_B\int_B|f(x)-f(y)|^p\,dμ(x)\,dμ(y)\\ &\quad\lesssimλ^pr_B^pϕ(B)^{γp}+\frac{λ^pr_B^p}{μ(B)}\mathop{\int_{ΛB}\int_{ΛB}}\limits_{\genfrac{}{}{}{2}{|f(x)-f(y)|}{ρ(x,y)ϕ(x,y)^γ}>λ}\frac{ϕ(x,y)^{γp}}{V(x,y)}\,dμ(x)\,dμ(y), \end{align*} where $V(x,y):=\mu (B(x,\rho (x,y)))$ and either $\varphi =\rho$ with $\varphi (B)=r_B$ and $\gamma \in (-\infty ,-\frac{\min \{q,D\}}{p}]$ or $\varphi =V$ with $\varphi (B)=\mu (B)$ and $\gamma \in (-\infty ,-\frac{q}{p\max \{q,d\}}]$. Here both ranges of $\gamma$ are sharp over the class of $q$-Poincaré spaces with the prescribed lower and upper dimensions, and the constant $\Lambda \in [1,\infty )$ and the implicit positive constant depend only on $p$, $\gamma$, and the geometric constants of $X$. We also obtain lower bounds for Sobolev energy and total variation in terms of the limit inferior of these functionals, for $\gamma$ in either the above negative ranges or $(0,\infty )$. Our proofs use the curve characterization of Poincaré spaces and an adaptive dyadic refinement argument based on medians.